This is a topic that I've entertained for several years now, posting about it intermittently (first on the HEMA Alliance forums, and more recently on scattered Facebook threads). This blog post is a modified version of one such Facebook post.
The Case for the Twerhaw as a Diagonal Cut
This is an explanation of why the standard Twerhaw (Zwerchhau, etc.), meaning the one that is taught first and used most, is/should be a short edge diagonal cut thrown at a shallow angle (say, 20-40 degrees) using the same mechanics and structure as the current popular interpretation (just with the plane of the cut rotated upward).
There is likewise an Unterhaw version along the same cutting lines, as given in Liechtenauer verse 51 (Twer zw dem pflueg / zw dem ochsen hart gefüg) and labeled the "Zwerch" by Joachim Meyer.
The horizontal Twerhaw (HTH), which has been the common interpretation for the past 20 years or so (I want to say Bart Walczak popularized it, but that was really before my time so I may be in error), is a valid technique in a few special situations but is merely a variation on this cut. The inappropriate use of the HTH as the default cut rather than the proper angled Twerhaw is the result of a confluence of bad translations and misinterpretation.
The case for the Twerhaw comes in six parts:
1 - The diagonal cut defends the hands better
2 - The diagonal cut binds better
3 - The diagonal cut leads to the follow-on devices in a more intuitive way
4 - The diagonal cut is used in Messer
5 - The diagonal cut arises from common fencing
6 - The diagonal cut matches the source material better
Number 6 is obviously the most important, but it's the longest section so the others might get lost if it came first.
1 - Defending the Hands
When used as a counter-cut, the HTH leaves the hands directly in the path of the incoming edge and often leads to at least incidental contact (which could be less incidental in a real encounter since swords are sharp and fingers are flimsy). There are tweaks that can be made to the strike to mitigate this risk to an extent, and there's an argument to be made that with the stickiness of sharps the edge would not threaten the defender's hand as much, but the fact is that with the correct Twerhaw these considerations evaporate—the gross mechanics of the Twer protect the hands automatically. By striking directly onto the opponent's sword, you line up your shield in such a way as to maximize its protective qualities, and the risk of hand hits is no greater than in, say, a Zornhaw.
2 - Creating a Superior Bind
Whereas the HTH travels in a path oblique to most strikes and often results in a messy edge-to-edge bind, the proper Twer strikes directly across the line of attack, typically resulting in a strong over-bind with your short edge pressing the opponent's flat. This not only gives you more control of the bind, but it also allows a greater degree of Fuhlen.
I suspect this is what the anonymous gloss in the Ms. 3227a means when it says that "And with this crossing-strike it is quite good to get on the sword with someone. And when one gets the other on the sword, no matter how it happened, the other may get away from him with great difficulty, and will be beaten by him to both sides with crossing-strikes." (Zabinski)
3 - Making the Devices Smoother
The proper Twer leads to the canonical follow-on devices much more smoothly than does the HTH. I leave the proof of this as an exercise for the reader. :P
Seriously, there are like a thousand and I don't have room to explain them all. They all work, and they usually work better. (It was attempting a counter to a counter to the Twer from Ringeck that convinced Jake Priddy, since it just sort of appeared spontaneously rather than having to be forced as was required by the HTH.)
4 - Twerhaw vs. Entrusthaw
The Entrusthaw (Shock or Anger Hew) is the equivalent hidden strike to the Twerhaw with the Messer—it uses the same verses in Lecküchner as the Twer does in Liechtenauer, and Andreas provides independent confirmation of this equivalence.
The foundational play of the Entrusthaw is breaking the guard Luginsland (Watchtower, i.e. vom Tag) starting from a low position comparable to an Alber. Performing this strike horizontally from a low guard is quite awkward and difficult, and there's simply no way to maintain a good speed and structure while doing it from any position. On the other hand, throwing it diagonally (by passing through the Bogen) is a smooth and powerful action.
The follow-on devices of the Entrusthaw likewise give no implication of a horizontal strike.
(Caveat: Jake Norwood has a Messer version of the HTH that maintains good structure, but I only like it when striking from Luginsland, not from the low wards.)
5 - Emergent from Common Fencing
Pseudo-Peter von Danzig's gloss tells us that the Zornhaw is "a simple peasant strike", and Ms. 3227a's gloss that no strike is more natural to a man in a rage; with training and art, however, this simple diagonal long-edge Oberhaw becomes one of Liechtenauer's five Hidden Strikes. The Twer is similar: teach someone that both edges are for striking and they will naturally start trying to throw short-edge Oberhawen (I did on my first day, and you probably did too). With training and art, these short-edge strikes become another of Liechtenauer's Hidden Strikes.
This is also why the only "named" strike that the gloss prepares his students to counter is the Twerhaw—because the "Twerhaw" he expects is the simple short-edge diagonal cut of the common fencer, and the proper responses are different than the typical long-edge cuts he just calls "Oberhawen".
6 - Matches the Source Materials
Most Liechtenauer texts make no indication of an angle, so someone coming to the treatises fresh would just assume a default diagonal angle for the Twer, or possibly take the statement from Ms. 3227a (23v-24r) to heart which says "Also know that there are only two strikes out of all the strikes, as they may always be called, that is the strike from above and the strike from below from both sides. These are the main strikes and the rudiment of all other strikes." (Zabinski) and from that surmise that the Twer, like all strikes, is a modified Oberhaw or Unterhaw from the side.
However, back in the early, steam-powered days of HEMA we didn't have a lot of resources, and most older/legacy interpretations (including HTH) derive from an amalgam of poor (by our standards) translations from Ms. 3227a, Ringeck, Talhoffer, and Meyer. In this case, Meyer has a horizontal strike which he calls the "Mittel oder Uberzwerchhauw" which corresponds to the HTH and which some people associated with the Twer (even though Meyer has a separate Zwerchhauw section which is a rising diagonal cut).
But the primary piece of evidence that people relied on was two passages from Ms. 3227a (27v) that hung entirely on the translation chosen for "Twer"; removing those interpretive decisions from the text gives the following:
"Here notice and know that for the entire sword no strike is as honest, as quick, as ready and as good as the [Twer]. And it goes crosswise to both sides, with both edges, the front and the back one, to all the openings, lower and upper ones. And everything that comes from the roof, that is the upper strikes, or whatever comes from above downward, all this a swordsman breaks and defends against with [Twer], who will perform them properly.
Thus he throws the sword well forward crosswise in front of the head, to whatever side he wants, in the same way as if he wanted to come to upper hangings or windings. But in the case of the [Twer] he turns the flats of the sword, the one upward and the other downward or below, and the edges [Twer] to the sides, the one to the right and the other to the left side." (Zabinski, modified by Chidester)
Many people at the time were inclined to translate Twer as "horizontal", which resulted in a sort of circular definition of the cut. This in turn lead to interpreting the description of edge and flat orientation as describing a horizontal cut.
However, Twer literally means "across" (or "athwart", which means "across the direction of travel"). This lead to the term "crosswise cut", which people who had horizontality on the brain were inclined to interpret as being synonymous with horizontal. If it is interpreted as meaning "across [the line]", or "across [the strike]", or "across [the sword]", then there's no implication of a horizontal strike anywhere, and the description of blade orientation merely signifies that it must be a shallower cut than the others.
(As a linguistic aside, the term Twer is a derivation of Quer, which means not only "across" and "crosswise" but also "aslant" and "diagonal". Zwer, of course, is a variant phonetic spelling of Twer. For more, see: http://woerterbuchnetz.de/cgi-bin/WBNetz/wbgui_py?sigle=Lexer&lemid=LT02846 )
I've also heard people cite the B branch of the Pseudo-Peter von Danzig gloss ("Jud Lew") as further evidence for the baseless HTH interpretation, as it indicates that the sword should be held with the flat on the shoulder before striking. I actually am glad they caught that one, since striking a diagonal cut from flat-shoulder-vom-Tag is even easier than striking a horizontal one (resting the flat on the shoulder shortens the distance the short edge has to travel, and a diagonal cut generally follows a shorter line to the target than a horizontal). But in any case, holding the sword in this way only changes starting blade orientation, it doesn't limit the lines you can cut.
As I said, this is an argument that I've had many times, but many people who put in time and test it out find it to produce generally superior results to the HTH and become converts. So even if nothing written here convinces you, then go out and give it a try anyway. The results may surprise you.
P.S. To address one more common complaint that I hear when this comes up, which I couldn't find a place to include above: no, this is not a Schilhaw. A Schilhaw, according to my (and others') interpretation, works rather like the one in this video, which is nothing like the Twerhaw mechanic: