The time modulation of the material parameters has created new opportunities in the synthesis of exotic metamaterial responses (Z. Deck-Léger, N. Chamanara, M. Skorobogatiy, M. G. Silveirinha and C. Caloz, “Uniform-velocity spacetime crystals,” Advanced Photonics, vol. 1, pp. 1 - 26, 2019, E. Galiffi, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huidobro, M. G. Silveirinha, R. Sapienza, A. Alù and J. B. Pendry, “Photonics of time-varying media,” Advanced Photonics, vol. 4, pp. 014002, 2022). Remarkably, it has been shown that time-varying material responses can be used to realize unidirectional guides and isolators, multifunctional nonreciprocal metasurfaces, and others, without an external magnetic bias. Recently, spacetime photonic crystals formed by isotropic materials with a travelling wave-type modulation were shown to reproduce the well-known moving medium coupling (P. Huidobro, M. G. Silveirinha, E. Galiffi and J. B. P. Pendry, “Homogenization Theory of Space-Time Metamaterials,” Physical Review Applied, vol. 16, pp. 014044-1-13, 2021). This type of modulation leads to a synthetic Fresnel drag effect such that the electromagnetic waves are dragged by the synthetic motion of the medium, in the same manner as they would be dragged by a moving material body. Moreover, in a recent work we have put forward a solution to realize the peculiar class of nonreciprocal Tellegen (axion) materials using a spacetime modulation (F. R. Prudêncio and M. G. Silveirinha, “Synthetic axion response with spacetime crystals,” arXiv:2209.03314, Sept. 2022). It was shown that spacetime modulations offer a new way to realize Tellegen-metamaterials with a giant axion-type response. Here, we will show that anisotropic spacetime crystals can be used to engineer arbitrary Hermitian bianisotropic nonreciprocal couplings in the long wavelength limit. We will show how by controlling in space and in time the permittivity and permeability optical axes and the optical indices, it may be possible to realize a (real-valued) magneto-electric tensor with an arbitrary structure and arbitrary symmetry. We will discuss how the symmetries of the magneto-electric tensor are linked to the symmetries of the corresponding static crystal, and provide examples where the magneto-electric tensor is anti-symmetric (moving medium coupling), a scalar (axion coupling) or a symmetric tensor with a zero trace (traceless Tellegen medium), which are the most general symmetry classes for Hermitian systems. It is envisioned that such nonreciprocal responses may enable interesting applications, such as, unidirectional transmission, electromagnetic isolation and exciting new forms of light-wave interactions.