Radical toe in once more

IME setting up bi directional speakers can be problematic. That back wave can be a killer if you don't treat the wall behind in in some fashion. I used dispersive material.  The success of crossing the speakers axis in front of the listener depends a lot on how much it is needed to reduce the effects of side wall/first reflections. It usually helps most when the speakers had to be placed close to the side walls. If side walls are not a problem then there really is little benefit other than changing the reflection pattern off the walls behind the speakers. I had Quad 65's for some years and it seemed to take forever to find the balance that worked best, butr I ended up with them pointed to just outside of my ears (no crossing over) and the rear pointing towards the room corners (I was 6ft off the rear wall). But, FWIW, it takes a long time (usually) to find the perfect placement for them to work, but do no minimize the need for rear wall treatment, Hell, I even tried Boston Ferns! Good luck.

Something rarely addressed is the effect of floor and, more importantly, ceiling reflections. Crossing the axis of your speakers in front of you might get a more diffused soundstage (for the better?) because of changing ceiling reflections. Whatever, it does make a difference if you listen closely.

Too much sound in your mind, not enuf in your ears I think. There is no right or wrong re toe in, it's what actually works for you. I would agree the aesthetically it might be a bit hard to think it could be right (for you) because few seem to do this. However, it is often done at audio shows to accommodate room/set up problems. 

You mentioned the tweeter off axis sounding better because it dials down the sound, but consider that the same 'off axis' can be achieved by pointing the speakers straight ahead. Much of what is achieved by toe in (or actually toe out for the really venturesome) is brought about by the change in sound from room reflections. Ditto diffusion/deadening wall treatment. 

One thing to think about, and much depends of speaker design, i.e. strength of the off axis signal, is what happens to the off axis signal when you change the speakers toe in. Pointed straight ahead you get a strong side wall reflection (if you have a side wall!) but toed in you change that reflection (but you knew that). Probably what you did not consider is what happens to the off axis signal on the other side. What you have by doing 'severe' toe in is to point that off axis signal off the wall between your speakers. Enter again the importance of the use (or not) of reflective/deadening/dispersive material between the speakers, not just  behind or to the side of, the speakers. For the really anal amongst us, don't fail to consider the 2d reflections points, they can be important as well. :-)

Too complex for the poor mind I think. It may be just simpler just to move your speakers about and listen to the results as you go. DON'T be in a hurry, beyond initial first impressions, the effects of moving speakers which are already established in a good place can be subtle and in my case affect the sound stage, especially image's third dimension, depth of image. No kidding, it probably took me over a year to set up my Quad 63'. And they don't have a strong off axis signal. My present speakers which are just forward firing boxes, not near so much!

Enuf for now, I'm even boring myself. :-)

Re hardcore science, well there are books (available on Amazon) on acoustics as they relate to audio systems, but I don't know how much they would help you in setting up your system.

I'm not familiar with the proper set up of bi-directional speakers. Forward firing boxes, panels, and electrostatics yes. I'm really not  familiar with  omni's or bi-directional boxes either, so I'm not a really good source of information for you.

The 'between the speakers" I referred to was from the inside of your forward facing drivers. I wasn't suggesting anything in particular, just that depending on the strength of the off axis signal that reflections off this wall could affect your systems sound (note I referred to 2d reflection points) but its contribution would be small(er) compared to the rear drivers reflecting off any of the walls sounding your speakers. 

But something I can rather confidently suggest you try is bringing your speakers out further into your room. This will give you better separation of the direct sound vs the sound of the rear wall, resulting in a clearer sound. At least 5 ft. Further won't hurt but be mindful of the change in bass response due to room nulls and nodes. 

FWIW.

FWIW, I think you would be wise(r) to follow Duke’s recommendations. He knows his speakers and their proper set up. He takes it seriously. I’ve been doing crossed axis’ for some years and I was tickled when I saw him using/recommending this set up. BTW, I experience no difficulty in reproducing an excellent soundstage. If its on the recording I get it! I see a lot of folks who think crossed axis’ compromise the sound stage. What I think they hear are 1st reflection points and think that is part of a normal sound stage. They think in-phase sound outside of the speaker edges is a norm and if they don’t hear it something with there stuff or set up is wrong. Not so, it is right! All one should hear outside the edges of their speakers is out of phase sound and to quote a test record producer "and it should come from all about the room". Note that Duke also has recommendations regarding the necessary distance of your speakers from a reflecting surface. Follow them! You’ll get much greater clarity. Having fun yet? :-)

Six feet is good! Do I assume correctly that they are also six feet forward of the wall behind them (front wall)? If so you're really good. FWIW this distance allows separation of the reflected  and the direct signal which give you a clearer sound of both.  

I was referring to the recorded sound. Sounds recorded in phase will appear between your two speakers, not outside of your speakers. Sounds recorded out of phase will appear from all about your room, but not focused between your two speakers.

Listen to a test record which has music recorded in phase and out of phase. When you listen to a cut recorded in phase you should hear nothing outside the speakers (in a well set up system which has no 1st reflection point reinforcement). Likewise when you play a cut with no in phase sound you will lose that focused sound we all love. (Recording engineers do play with phase when recording music which can give an enhanced sound stage.) Getting your system set up properly entails getting reproduced sounds, either in phase or out of phase, as put down in the disc. The Sheffield/XLO Test & Burn-in Disc has such cuts with an explanation. With this and some other cuts it can be quite helpful in setting up your system. 

This has absolutely nothing to do with the performance of your speakers with which you seem to be concerned. That is an entirely different issue. 

Results do not surprise me at all. I have similar issues in my present main room but, in another room using 'crap' stuff I get the out of phase sounds from the rear and sides of the room as well. I conclude this has more to do with the room and set up than anything else. But I'm OK with it. If I really wanted a room full of sound I'd probably cheap out and just use an old fashioned Hafler set up but it has too many distracting problems in trying to get it set up, but as I recall (and I could be wrong) it relied on out of phase sound coming from 2 additional speakers in the rear of the room. An over simplified version of 4 channel sound requiring only an attenuator for the rear channels. I think you'd hate it! :-)