Radical toe in once more

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.

I have ESP loudspeakers in one of my systems.  These speakers are specifically designed for a 45 degree toe in.  In my summer home they are setup in a spare bedroom 10x13x8, technically too small of a room according to the instruction manual.  The speakers are 42 inches from the front wall, measured from the furthest edge of the front baffle.  My listening position is approximately 7 feet away.  In this setup, I ended up reducing the toe in to around 35 degrees so the intersection occurs just at the front of my chair.  The resulting soundstage is much to my liking.  Out of my four systems this is the only one that consistently delivers the “you are there” listening experience versus the “they are here” experience.  It is more of a concert like effect and I like that.

This room is a good sounding room to start with and is treated with two diffusion panels, two tube traps, and several absorption panels strategically located by listening.

I don’t think this degree of toe in will work with most speakers or rooms.  However, experimentation is the only way to find out.

Is there an "inherent" problem with radical toe in, when the main sound from the front drivers cross in front of the listener, instead of the more conventional setup where the crossing point is behind the listener - and if so, what?

@o_holter Toe-in helps you to minimize side wall reflections, which are interpreted by the ear as harshness. I don't think there is a 'more conventional setup where the crossing point is behind' for this speaker. Generally if there are side walls nearby, that might not work out so well.

Yesterday I bought Endresen and Wesseltoft: Out here, in there, on 2 x LP 45 rpm (Jazzland records). It sounded so good that my wife played the whole thing once more (doesnt happen often). So I think we're nearly there. The setup is much like @rhljazz described. Ca. 40 degrees toe-in. I don't hear any front wave disturbance with the female vocal. The treble is a bit tamed, which is OK in my case. In a sense, this whole thing is like going back to start. Following the speaker designer's advice, with some minor adjustments from there. It will be interesting what my audio friends think, since they preferred conventional toe-in, some years ago. Anyway, it has been an interesting experiment, so thanks again for all advice.

Finding the best sounding firing axis angle is something that really needs to be determined by experimentation (as well as overall speaker positioning!) 

Are the frequency response measurement plots available for your speakers that show different amounts of on and off axis firing angles?   Ideally you would want to aim them such that the response of the top end gradually tapers off for the most pleasing sound.  

Get them in the best general position from room boundaries to give you the best sounding bass and aim them straight ahead, perpendicular from the wall. 

Listen for a few days with your demo songs and take notes, good, bad or other.  Listen for tonal balance, brightness, sound stage, center image, etc.   

Then aim them with some toe in- say 10 degrees or so (maybe ai inch from the inside front corner with tape marks.  Listen again.

Repeat the process inch by inch until you get to the extreme toe angle that aims a foot or so in front of you.  

Review your notes and put them in the best sounding spot!  

Lots of work but that is how you get the best sound out of your speakers