Regular self-dual and self-Petrie-dual maps of arbitrary valency

The main result of D. Archdeacon, M. Conder and J. \v{S}ir'a\v{n} [Trans. Amer. Math. Soc. 366 (2014) 8, 4491-4512] implies existence of a regular, self-dual and self-Petrie dual map of any given even valency. In this paper we extend this result to any odd valency $\ge 5$. This is done by algebraic number theory and maps defined on the groups $\mathrm{P}\mathrm{S}\mathrm{L}(2,p)$ in the case of odd prime valency $\ge 5$ and valency $9$, and by coverings for the remaining odd valencies.