  Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda,
l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb
Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and
$\lambda=(\lambda_1, \dots, \lambda_n)$ is a partition. G\'omez-Ullate et al
showed that for a special class of partitions the corresponding polynomials are
orthogonal and dense among all polynomials with certain inner product, but in
contrast to the usual case have some degrees missing (so called exceptional
orthogonal polynomials). We generalise their results to all partitions by
considering complex contours of integration and non-positive Hermitian
products. The corresponding polynomials are orthogonal and dense in a
finite-codimensional subspace of $\mathbb C[x]$ satisfying certain
quasi-invariance conditions. A Laurent version of exceptional orthogonal
polynomials, related to monodromy-free trigonometric Schr\"odinger operators,
is also presented.
