The authors discuss two methods of symmetrizing Markov processes. The first method applies to some Lévy processes X on a compact Abelian group G, with \(\alpha\)-potential density \(u^{\alpha}(x,y)\). A condition is given which guarantees that \(v^{\alpha}(x,y)=u^{\alpha}(x,y)+u^{\alpha}(y,x)\) will be the \(\alpha\)-potential density of a symmetric Lévy process on G, which in turn implies that the original non-symmetric process X must satisfy the \(\alpha\)-maximum principle and Hunt's hypothesis (H) (``semipolar sets are polar''). The second method applies to Hunt processes X having a dual, where the dual transition operators \(P_ t\) and \(\hat P_ s\) commute. It is shown that the associated ``energy space'', obtained by completing the space of one-potentials U 1f with \(f\in L\) 2 relative to a certain inner product, is the Dirichlet space of a symmetric process Y, and that semipolar sets for X are polar for Y.