The higher-order tensor renormalization group is a tensor-network method providing estimates for the partition function and thermodynamical observables of classical and quantum systems in thermal equilib-rium. At every step of the iterative blocking procedure, the coarse-grid tensor is truncated to keep the tensor dimension under control. For a consistent tensor blocking procedure, it is crucial that the forward and back-ward tensor modes are projected on the same lower-dimensional subspaces. In this paper we present two methods, the SuperQ and the iterative SuperQ method, to construct tensor truncations that reduce or even minimize the local approximation errors, while satisfying this constraint. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/). Funded by SCOAP3.