  We give an almost-complete description of orthogonal matrices $M$ of order
$n$ that "rotate a non-negligible fraction of the Boolean hypercube
$C_n=\{-1,1\}^n$ onto itself," in the sense that $$P_{x\in C_n}(Mx\in C_n) \ge
n^{-C},\mbox{ for some positive constant } C,$$ where $x$ is sampled uniformly
over $C_n$. In particular, we show that such matrices $M$ must be very close to
products of permutation and reflection matrices. This result is a step toward
characterizing those orthogonal and unitary matrices with large permanents, a
question with applications to linear-optical quantum computing.
