The behavior of sloshing eigenvalues and eigenfunctions is studied for vertical cylindrical containers that have circular walls and constant (possibly infinite) depth. The effect of breaking the axial symmetry due to the presence of radial baffles is analyzed. It occurs that the lowest eigenvalues are substantially smaller for containers with baffles going throughout the depth; moreover, all eigenvalues are simple in this case. On the other hand, the lowest eigenvalue has multiplicity two in the absence of baffle. It is shown how these properties affect the location of maxima and minima of the free surface elevation and the location of its nodes.