Consider a Cayley graph \(\text{Cay}(G,S)\) of a finite group \(G\) generated by a set \(S=S^{-1}=\{s_0,\ldots,s_{|S|-1}\}\) where \(1\notin S\). A bijection \(\omega:G\to G\) is called a complete rotation of the graph if \(\omega(1)=1\) and \(\omega(xs_i)=\omega(x)s_{i+1}\) for all \(x\in G\) and all \(i\in{\mathbb{Z}}_{|S|}\). A graph is rotational if it can be represented as a Cayley graph that admits a complete rotation. This quasi-expository paper presents various necessary and/or sufficient conditions for the existence of complete rotations. For example, \(K_n\) is rotational if and only if \(n\) is a power of a prime (Theorem 2.2). Given a finite group \(G\) generated by a set \(S\), the following are equivalent: (i) \(\text{Cay}(G,S)\) admits a complete rotation; (ii) for any presentation \(\langle S|R\rangle\) of \(G\), the free group \(F(S)\) admits an automorphism that fixes setwise the normalizer of \(R\) in \(F(S)\) and also induces a cyclic permutation of \(S\); (iii) there exists a presentation \(\langle S|R\rangle\) of \(G\) such that \(F(S)\) admits an automorphism that fixes \(R\) and also induces a cyclic permutation of \(S\) (Corollary 3.1).