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I'm making some additions to my stereo system and I'll try to explain
this as concise as possible. I have two additional speakers which I'd
like to connect to the receiver in the living room. The room is about
15 X 22 feet. The existing two speakers are situated against an inside
wall facing the windows. They are properly phased and sound very good.
One of these existing speakers (left side speaker A system), is very
slightly caddy corner so that it projects into the room. I would like
to connect the additional left side counterpart on the opposite window
wall so that it's speaker, (proposed speaker B system) is doing the
same thing from the corner it's in. The new right speaker from proposed
system B will essentially facing it's counterpart from speaker A
system,separated by about 15 feet. What I'm trying to accomplish is
more sound filling the room from more directions and hopefully
simulating a sort of pseudo "surround" type of effect. But I'm not sure
about how this is going to work out. Perhaps this is a mistake, so I
thought I'd ask.

If all four speakers are facing one another and are in phase, and by
this I mean before connecting to the receiver confirming that a small
battery makes all four cones move in the same direction, connected like
this what happens when the two lefts for instance are outputting the
same signal. With the cones facing each other, will the projected sound
buck and effectively try to cancel? Or should both cones in proposed B
system be in phase as a pair but out of phase with respect to A system?
Or should I forget the whole thing and just stick with my two existing
A speakers? I hope I've explained this well. Thanks for any advice.

In the situation you've suggested - if the two left-channel speakers
are in phase, then their outputs will reinforce. You'll be doing
something like moving the location of the left output to a point
located half-way between the two left-channel speakers. Ditto on the
right side. This may or may not give you an effect you like... it may
"open up" the sound, but at the cost of creating a less-realistic
"sound stage" stereo image between the left front and right front
speakers.

You might want to experiment with a "phase difference" approach for
your rear speakers... something akin to the old Dynaquad system. The
simplest way to do this, is to wire the "-" inputs of the two
rear-channel speakers together (and not back to the amp). Run the "+"
inputs of the left rear and right rear speakers to the "+" terminals
of the left and right amplifer outputs, respectively. You may wish to
include a rheostat or pad somewhere in this wiring, so you can reduce
the (relative) sound level from the rear speakers.

In this setup, the parts of the musical signal which are in phase and
at the same volume, will be played only from the front speakers, and
will appear to be "dead center". As an instrument or singer moves to
the left or right, and the amplitude or phase of their music is
different between the two channels, the rear speakers will begin to
play it (in addition to one or both of the front speakers). Sounds
which are very different in phase between the two channels
(e.g. reflected "hall ambience" sounds) will be played most strongly
from the rear speakers.

This is a simple and inexpensive setup to create, and it can sound
surprisingly good. Compared to the "two left and two right" layout
you suggested, it tends to leave you with a more realistic left/right
stereo image.

If I recall correctly, the original Dynaquad setup works a bit
differently. It requires four identical speakers (the above system
works OK with two pairs that can be different from one
another)... left, right, rear, and center... and the wiring is a bit
more complex. Similar idea, though.