Neural compression has brought tremendous progress in designing lossy compressors with good rate-distortion (RD) performance at low complexity. Thus far, neural compression design involves transforming the source to a latent vector, which is then rounded to integers and entropy coded. While this approach has been shown to be optimal on a few specific sources, we show that it can be highly sub-optimal on synthetic sources whose intrinsic dimensionality is greater than one. With integer rounding in the latent space, the quantization regions induced by neural transformations, remain square-like and fail to match those of optimal vector quantization. We demonstrate that this phenomenon is due to the choice of scalar quantization in the latent space, and not the transform design. By employing lattice quantization instead, we propose Lattice Transform Coding (LTC) and show that it approximately recovers optimal vector quantization at reasonable complexity. On real-world sources, LTC improves upon standard neural compressors. LTC also provides a framework that can integrate structurally (near) optimal information-theoretic designs into lossy compression; examples include block coding, which yields coding gain over optimal one-shot coding and approaches the asymptotically-achievable rate-distortion function, as well as nested lattice quantization for low complexity fixed-rate coding.