Theoretical studies of angular momentum transport suggest that isolated stellar-mass black holes are born with negligible dimensionless spin magnitudes $\chi \lesssim 0.01$. However, recent gravitational-wave observations indicate $\gtrsim 15\%$ of binary black hole systems contain at least one black hole with a non-negligible spin magnitude. One explanation is that the first-born black hole spins up the stellar core of what will become the second-born black hole through tidal interactions. Typically, the second-born black hole is the ``secondary'' (less-massive) black hole, though, it may become the ``primary'' (more-massive) black hole through a process known as mass-ratio reversal. We investigate this hypothesis by analysing data from the third gravitational-wave transient catalog (GWTC-3) using a ``single-spin'' framework in which only one black hole may spin in any given binary. Given this assumption, we show that at least $28\%$ (90% credibility) of the LIGO--Virgo--KAGRA binaries contain a primary with significant spin, possibly indicative of mass-ratio reversal. We find no evidence for binaries that contain a secondary with significant spin. However, the single-spin framework is moderately disfavoured (natural log Bayes factor $\ln B = 3.1$) when compared to a model that allows both black holes to spin. If future studies can firmly establish that most merging binaries contain two spinning black holes, it may call into question our understanding of formation mechanisms for binary black holes or the efficiency of angular momentum transport in black hole progenitors.