Hm. Sorry to have wasted your time. I'd picked up on this bit of the OP

... since A and E are restricted to only two values between them, they must consume those two values;

... and hung my hat on it, but that obviously doesn't apply in the same way to the two examples above.

Question: Would this example also decompose into the (same?) two groups as the above?

my @grid = (
    [ 0, 1, 0, 1, 0, 0, ],
    [ 0, 0, 0, 1, 1, 0, ],
    [ 0, 1, 0, 0, 1, 0, ],
    [ 1, 0, 0, 0, 0, 1, ],
    [ 1, 0, 1, 0, 0, 0, ],
    [ 1, 0, 0, 0, 0, 1, ],
);
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