  A class of unital qubit maps displaying diagonal unitary and orthogonal
symmetries is analyzed. Such maps already found a lot applications in quantum
information theory. We provide a complete characterization of this class of
maps showing intricate relation between positivity, operator Schwarz
inequality, and complete positivity. Finally, it is shown how to generalize the
entire picture beyond unital case (so called generalized Schwarz maps).
Interestingly, the first example of Schwarz but not completely positive map
found by Choi belongs to our class. As a case study we provide a full
characterization of Pauli maps. Our analysis leads to generalization of seminal
Fujiwara-Algoet conditions for Pauli quantum channels.
