  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 halos, 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 $P_{\rm 1.4 \, GHz}\, M_{500}$
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 halos (the null hypothesis that the
clusters belong to the same group cannot be rejected). However, the data
suggests that the absence of radio halos 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.
