Outbreaks of hand, foot and mouth disease have been documented in Japan since 1963. This disease is primarily caused by the two closely related serotypes of Enterovirus A71 (EV-A71) and Coxsackievirus A16 (CV-A16). Here, we analyse Japanese virologic and syndromic surveillance time-series data from 1982 to 2015. As in some other countries in the Asia Pacific region, EV-A71 in Japan has a 3 year cyclical component, whereas CV-A16 is predominantly annual. We observe empirical signatures of an inhibitory interaction between the serotypes; virologic lines of evidence suggest they may indeed interact immunologically. We fit the time series to mechanistic epidemiological models: as a first-order effect, we find the data consistent with single-serotype susceptible-infected-recovered dynamics. We then extend the modelling to incorporate an inhibitory interaction between serotypes. Our results suggest the existence of a transient cross-protection and possible asymmetry in its strength such that CV-A16 serves as a stronger forcing on EV-A71. Allowing for asymmetry yields accurate out-of-sample predictions and the directionality of this effect is consistent with the virologic literature. Confirmation of these hypothesized interactions would have important implications for understanding enterovirus epidemiology and informing vaccine development. Our results highlight the general implication that even subtle interactions could have qualitative impacts on epidemic dynamics and predictability.