The author studies the set \(Z(F)\) of rational maps of the Riemann sphere which commute (under composition of maps) with a given finite Blaschke product of degree \(n\), \(F(z):= a_0 \prod_{i=1}^n \frac{z-\bar a_i} {1-a_i z}\) (where \(|a_0|=1\) and \(|a_i| 3\); or (ii) \(n=3\) and \(a_0 \neq 1\); or (iii) \(n=2\) and \(a_0 \neq 1, -1\) and there exists a fixed point \(x\) of \(F\) on the unit circle such that \(F'(x)\neq 2\). Examples are given to show that the additional hypotheses are needed in the cases \(n=2\) or \(3\). More generally, what is actually shown as an intermediate step in those cases is that under hypothesis (H), the non-trivial part of \(Z(F)\) has to be included in the maps of the form \(\frac 1{F^k}\), \(k \in \mathbb N\). The idea of the proofs is first to exploit the fact that the two maps that commute have a common Julia set, and that \(G\) preserves the unit circle; in some cases \(G^2\) is a Blaschke product, and the analogous result of \textit{C. Arteaga} [Ergodic Theory Dyn. Syst. 19, No. 3, 549-552 (1999; Zbl 0934.30029)] (where the case of \(G\) being a Blaschke product was studied) can be applied. In general, dynamical arguments coming from \textit{G. M. Levin} [Math. Notes 48, No. 5, 1126-1131 (1990); translation from Mat. Zametki 48, No. 5, 72-79 (1990; Zbl 0724.30021)] show that for some integers \(i\), \(j\), \(G^i=F^j\); and then hypothesis (H) provides enough common fixed points of the maps (in the high degree cases) to conjugate them to powers of \(z\) of the same respective degrees; lastly, arithmetic arguments will provide relations between the degrees.