In this paper, we first introduce a generalized modeling and analysis framework to explore the fundamental interactions between user association, coverage probability and link rate in a millimeter wave (mmWave) heterogeneous cellular network (HetNet) in which there are multiple tiers of the ultra-high-frequency (UHF) macrocell and small cell base stations (BSs) and a single tier of mmWave small cell BSs. A generalized user association scheme that can cover many path-loss-based user association schemes is proposed and its related probabilistic properties that facilitate the derivations of the coverage probability and link rate are derived. The derived general expressions of the coverage and link rate not only shed light on how to design user association functions in order to maximize the coverage and link rate but also show that it is impossible to devise a user association scheme that maximizes the coverage and link rate at the same time. Namely, there exists a fundamental tradeoff between the coverage and link rate in mmWave HetNets with distinct bandwidths in the UHF and mmWave bands while a user is associating with a BS. We characterize the coverage-optimal and rate-optimal user association schemes and numerically validate their performances and show the coverage-rate tradeoff problem.