  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. Asymmetries have
been detected in the rotational multiplets of several red giants. This is
unexpected since all the red giants whose rotation 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. 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. We successfully apply our method to the asymmetrical $l=2$
multiplets of the Kepler young red giant KIC7341231 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. 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.
