We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in
$S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences
are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb
R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$
by $(Z_2)^{k+1}$-symmetric convex fans.