  Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph
$\Gamma^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$
(projections of rank $k$) and two subspaces are connected by an edge in this
graph if they are compatible and adjacent (the corresponding rank-$k$
projections commute and their difference is an operator of rank $2$). Our main
result is the following: if $\dim H\ne 2k$, then every automorphism of
$\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator; if
$\dim H=2k\ge 6$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced
by a unitary or anti-unitary operator or it is the composition of such an
automorphism and the orthocomplementary map. For the case when $\dim H=2k=4$
the statement fails. To prove this statement we compare geodesics of length two
in ortho-Grassmann graphs and characterise compatibility (commutativity) in
terms of geodesics in Grassmann and ortho-Grassmann graphs. At the end, we
extend this result on generalised ortho-Grassmann graphs associated to
conjugacy classes of finite-rank self-adjoint operators.
