For fixed real or complex matrices A and B, the well known von Neumann trace inequality identifies the maximum of ⎸tr(U AV B) ⎸, as U and V range over the unitary group, the maximum being a bilinear expression in the singular values of A y B. This paper establishes the analogue of this inequality for real matrices A and B when U and V range over the proper (real) orthogonal group. The maximum is again a bilinear expression in the singular values but there is a subtracted term when A and B have determinants of opposite sign.