If A is a real 2n×2n positive definite matrix, then there exists a symplectic matrix M such that MTAM=diag(D,D), where D is a positive diagonal matrix with diagonal entries d1(A)⩽⋯⩽dn(A). We prove a maxmin principle for dk(A) akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality di+j−1(A+B)⩾di(A)+dj(B).