That was it, 100%! I ended up drawing out each combination of basis vectors. Visually, it becomes obvious which one's can't be rotated to. In case anyone reading this thread in the future wants to know, valid combinations are:

+ + +
+ - -
- - +
- + -

I have one last question which is kind of unrelated (but kind of related). when doing a polar decomposition and adding the matrix to its inverse transpose, is that to be interpreted as:

Q = 0.5 * (Q + Inverse(Transpose(Q)))

Or as:

Q = 0.5 * (Q + Transpose(Inverse(Q)))

The wording on this has always confused me. Are we transposing the inverse, or taking the inverse of the transpose?

Interesting question. I would argue it doesn't matter. Since

( (a)^n )^m = a^(n*m) = a^(m*n) = ( (a)^m )^n