Exactly (and firstly, may I apologise for leaving the word 'sideways' in my original post; it should have been edited out).
I don't see this as possible at all... I take it you mean the bino is now oriented such that when pointed toward the horizon, one objective lies directly above the other, effectively having the instrument lying on what would currently be considered the side.
Starfields don't have a 'proper orientation' (if so, they'd be 'upside down' in an absolute sense in the southern hemisphere) so it matters not a jot what their actual orientation is when you observe them - OK, it might be unfamiliar, and require star atlases to be rotated, but it wouldn't be 'wrong'. I accept that it seems less 'right' to do this with a binocular than it does with a telescope because we normally use bins with their OG's parallel to the horizon, but whilst this is a necessity in terrestrial use it is irrelevant for astronomy (asymmetric binoculars are an example of this).
There can't be many knowledgable 'scope users who haven't thought, "I'd love a Springfield mount", because having the fixed observing position is just SO convenient, but they're complicated and probably not compatible with low magnification/wide field aspirations. Designing a Springfield-mount binocular would be a nightmare, but if you use large bins and want your observing position to be as static as possible - and are prepared to compromise by accepting that you could enjoy a 'Springfield effect' for objects at ANY declination, but ONLY across a small-ish range of RA - then surely this would be worth going for. I envisage a seated observer, using a bottom-mounted Mr. Bill design, sitting side-on to whatever he is observing (we already do this with Newtonian telescopes), with his head ALWAYS held in its natural position (upright and with the eyes looking parallel to the ground), observing an object at ANY declination through an arc of 180 degrees, with only small movements of the head being required to compensate for changes in declination. As I said in my original post, if the (principal axes of the) OG's are around 9" apart, and the instrument is bottom-mounted, and the 1.25" diagonals have a small footprint, then for diagonals arms whose diagonals have their centres 4.5" apart (as per Mr Bill's layout in post #1), it is ALWAYS possible to orient the diagonal arms so that horizontally-aligned eyepieces are presented to the observer with the correct IPD (you really have to make a life-size scale model of the system to see the complex geometry that's going on when you swing the diagonal arms around - it's not something you can visualize). The eyepiece position (compared to a fixed datum) DOES move a little (both vertically and horizontally) for changes in declination, but can be compensated for - even if the observer is seated in a fixed-height chair - by slight changes in posture. The observer only ever needs to significantly alter the position of his head when he gets up to move his seat so that he can observe another object which has an appreciably different RA.
If you think about it, when forum members are discussing the relative merits of straight-through, 45-degree or 90-degree eyepiece inclinations (and bemoaning the fact that the preferred 90-degree option usually comes with reduced-aperture problems) then we're really talking about keeping head position as natural as possible, to reduce fatigue; the bottom-mounted Mr. Bill instrument achieves this and all the design work has been done.