Context. The Kepler space mission has made it possible to measure the rotational splittings of mixed modes in red giants, thereby providing an unprecedented opportunity to probe the internal rotation of these stars.Aims. Asymmetries have been detected in the rotational multiplets of several red giants. This is unexpected since all the red giants whose rotation profiles have been measured thus far are found to rotate slowly, and low rotation, in principle, produces symmetrical multiplets. Our aim here is to explain these asymmetries and find a way of exploiting them to probe the internal rotation of red giants.Methods. We show that in the cases where asymmetrical multiplets were detected, near-degeneracy effects are expected to occur, because of the combined effects of rotation and mode mixing. Such effects have not been taken into account so far. By using both perturbative and non-perturbative approaches, we show that near-degeneracy effects produce multiplet asymmetries that are very similar to the observations. We then propose and validate a method based on the perturbative approach to probe the internal rotation of red giants using multiplet asymmetries.Results. We successfully apply our method to the asymmetrical l = 2 multiplets of the Kepler young red giant KIC 7341231 and obtain precise estimates of its mean rotation in the core and the envelope. The observed asymmetries are reproduced with a good statistical agreement, which confirms that near-degeneracy effects are very likely the cause of the detected multiplet asymmetries.Conclusions. We expect near-degeneracy effects to be important for l = 2 mixed modes all along the red giant branch (RGB). For l = 1 modes, these effects can be neglected only at the base of the RGB. They must therefore be taken into account when interpreting rotational splittings and as shown here, they can bring valuable information about the internal rotation of red giants.