Pairs of radio relics are believed to form during cluster mergers, and are best observed when the merger occurs in the plane of the sky. Mergers can also produce radio haloes, through complex processes likely linked to turbulent re-acceleration of cosmic ray electrons. However, only some clusters with double relics also show a radio halo. Here, we present a novel method to derive upper limits on the radio halo emission, and analyse archival X-ray Chandra data, as well as galaxy velocity dispersions and lensing data, in order to understand the key parameter that switches on radio halo emission. We place upper limits on the halo power below the P1.4 GHz-M500 correlation for some clusters, confirming that clusters with double relics have different radio properties. Computing X-ray morphological indicators, we find that clusters with double relics are associated with the most disturbed clusters. We also investigate the role of different mass-ratios and time-since-merger. Data do not indicate that the merger mass-ratio has an impact on the presence or absence of radio haloes (the null hypothesis that the clusters belong to the same group cannot be rejected). However, the data suggest that the absence of radio haloes could be associated with early and late mergers, but the sample is too small to perform a statistical test. Our study is limited by the small number of clusters with double relics. Future surveys with LOFAR, ASKAP, MeerKat and SKA will provide larger samples to better address this issue.