  For each link L in S^3 and every quantum grading j, we construct a stable
homotopy type X^j_o(L) whose cohomology recovers Ozsvath-Rasmussen-Szabo's odd
Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(L), following a construction of
Lawson-Lipshitz-Sarkar of the even Khovanov stable homotopy type. Furthermore,
the odd Khovanov homotopy type carries a Z/2 action whose fixed point set is a
desuspension of the even Khovanov homotopy type. We also construct a Z/2 action
on an even Khovanov homotopy type, with fixed point set a desuspension of
X^j_o(L).
