  We investigate permutation-invariant continuous variable quantum states and
their covariance matrices. We provide a complete characterization of the latter
with respect to permutation-invariance, exchangeability and representing convex
combinations of tensor power states. On the level of the respective density
operators this leads to necessary criteria for all these properties which
become necessary and sufficient for Gaussian states. For these we use the
derived results to provide de Finetti-type theorems for various distance
measures.
