The main result of D. Archdeacon, M. Conder and J.Širáň [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 ≥ 5. This is done by algebraic number theory and maps defined on the groups PSL(2, p) in the case of odd prime valency ≥ 5 and valency 9, and by coverings for the remaining odd valencies.

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