We determine the Hausdorff dimension of the set of k-multiple points for a symmetric operator semistable Levy process $$X=\{X(t), t\in {\mathbb {R}}_+\}$$ in terms of the eigenvalues of its stability exponent. We also give a necessary and sufficient condition for the existence of k-multiple points. Our results extend to all $$k\ge 2$$ the recent work (Luks and Xiao in J Theor Probab 30(1):297–325, 2017) where the set of double points $$(k = 2)$$ was studied in the symmetric operator stable case.