Eulerian and Hamiltonian Graphs · Challenge problem

The pentagonal prism graph has two disjoint $5$-cycles (call them the "top" $0,1,2,3,4$ and the "bottom" $5,6,7,8,9$) plus five rungs $i\,(i+5)$ for $i=0,\dots,4$, so every vertex has degree $3$. Check that for EVERY pair of non-adjacent vertices in this $10$-vertex graph, Ore's theorem's degree-sum condition fails. Given that, how many distinct Hamiltonian cycles does the prism actually have?