Let M be a surface of revolution; M is topologically a sphere or torus. The rotation group O(2) acts on M by isometries and also on the eigenspaces E(\(\lambda)\) for the Laplacian D. Each irreducible representation of dimension 2 is characterized by a winding number \(k\in {\mathbb{N}}\). The author shows eigenfunctions in an irreducible component of E(\(\lambda)\) have the form \[ (*)\quad f(t)\cdot \cos \{k\cdot (\phi - \phi_ 0)\} \] where \(\tau\) is the arc length parameter on the meridians and \(\phi\) is the rotation angle on the parallels; the eigenfunctions of the form (*) give a basis for each E(\(\lambda)\). Let \(E(\lambda,k)\subseteq E(\lambda)\) is the subspace corresponding to winding number k; the topology of M controls the multiplicities. If M is homeomorphic to a sphere, \[ \dim E(\lambda,0)\leq 1\quad and\quad \dim E(\lambda,k)\in \{0,2\}\quad for\quad k\geq 1; \] if M is homeomorphic to a torus, \[ \dim E(\lambda,0)\leq 2\quad and\quad \dim E(\lambda,k)\in \{0,2,4\}\quad for\quad k\geq 1. \] The author also discusses the relevant nodal domains and bounds for eigenvalues. The paper uses results of \textit{J. Brüning} and \textit{E. Heintze} [Invent. Math. 50, 169-203 (1979; Zbl 0392.58015); Math. Ann. 269, 95-101 (1984; Zbl 0553.53028)].