We develop a higher order generalization of the LQ decomposition and show that this decomposition plays an important role in likelihood-based estimation and testing for separable, or Kronecker structured, covariance models, such as the multilinear normal model. This role is analogous to that of the LQ decomposition in likelihood inference for the multivariate normal model. Additionally, this higher order LQ decomposition can be used to construct an alternative version of the popular higher order singular value decomposition for tensor-valued data. We also develop a novel generalization of the polar decomposition to tensor-valued data.