  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 ${\rm PSL}(2,p)$ in the case of odd prime
valency $\ge 5$ and valency $9$, and by coverings for the remaining odd
valencies.
