The emergence of ratchet effects in two-dimensional materials is strongly correlated with the introduction of asymmetry into the system. In general, dual-grating-gate structures forming lateral asymmetric superlattices provide a suitable platform for studying this phenomenon. Here, we report on the observation of ratchet effects in $\mathrm{Hg}\mathrm{Te}$-based dual-grating-gate structures hosting different band-structure properties. Applying polarized terahertz laser radiation we detect linear and polarization-independent ratchets, as well as a radiation-helicity-driven circular ratchet effect. Studying the ratchet effect in devices made of quantum wells (QWs) of different thickness, we observe that the magnitude of the signal substantially increases with decreasing QW width with a maximum value for devices made of QWs of critical thickness hosting Dirac fermions. Furthermore, sweeping the gate voltage amplitude we observe sign-alternating oscillations for gate voltages corresponding to $p$-type conductivity. The amplitude of the oscillations is more than two orders of magnitude larger than the signal for $n$-type conducting QWs. The oscillations and the signal enhancement is shown to be caused by the complex valence band structure of $\mathrm{Hg}\mathrm{Te}$-based QWs. These peculiar features of the ratchet currents make these materials an ideal platform for the development of terahertz applications.