A random measure ξ \xi defined on some measurable space ( S , S ) (S,\mathcal {S}) is said to be symmetrically distributed with respect to some fixed measure ω \omega on S S , if the distribution of ( ξ A 1 , ⋯ , ξ A k ) (\xi {A_1}, \cdots ,\xi {A_k}) for k ∈ N k \in N and disjoint A 1 , ⋯ , A k ∈ S {A_1}, \cdots ,{A_k} \in \mathcal {S} only depends on ( ω A 1 , ⋯ , ω A k ) (\omega {A_1}, \cdots ,\omega {A_k}) . The first purpose of the present paper is to extend to such random measures (and then even improve) the results on convergence in distribution and almost surely, previously given for random processes on the line with interchangeable increments, and further to give a new proof of the basic canonical representation. The second purpose is to extend a well-known theorem of Slivnyak by proving that the symmetrically distributed random measures may be characterized by a simple invariance property of the corresponding Palm distributions.