The Calder on operator S is the sum of the the Hardy averaging operator and its adjoint. The weights w for which S is bounded on L p (w) are the Calder on weights of the class Cp. We give a new characterization of the weights in Cp by a single condition which allows us to see thatCp is the class of Muckenhoupt weights associated to a maximal operator dened through a basis in (0;1). The same condition characterizes the weighted weak- type inequalities for 1 < p < 1, but that the weights for the strong type and the weak type dier for p = 1. We also prove that the weights in Cp do not behave like the usual Ap weights with respect to some properties and, in particular, we answer an open question on extrapolation for Muckenhoupt bases without the openness property.