Interplanetary coronal mass ejections (ICMEs) are defined as "coherent" if they are capable of responding to external perturbations in a collective manner. This implies that information must be able to propagate across ICME structures, and if this is not the case, single-point in situ measurements cannot be considered as indicative of global ICME properties. Here, we investigate the role of Alfvénic fluctuations (AFs) as mediators of ICME coherence. We consider multipoint magnetic field and plasma measurements of 10 ICMEs observed by the ACE and Wind spacecraft at 1 au at longitudinal separations of 0.5°–0.7°. For each event, we analyze the Alfvénicity in terms of the residual energy and cross helicity of fluctuations, and the coherence in terms of the magnetic correlation between Wind and ACE. We find that ∼65% and 90% of ICME sheaths and magnetic ejecta (MEs), respectively, present extended AFs covering at least 20% of the structure. Cross helicity suggests AFs of solar and interplanetary origin may coexist in the ICME population at 1 au. AFs are mainly concentrated downstream of shocks and in the back of MEs. The magnetic field is poorly correlated within sheaths, while the correlation decreases from the front to the back of the MEs for most magnetic field components. AFs are also associated with lower magnetic field correlations. This suggests either that ICME coherence is not mediated by Alfvén waves, implying that the coherence scale may be smaller than previously predicted, or that the magnetic field correlation is not a measure of coherence.