It is generally assumed that the parent-twin interface is a fully invariant plane. The calculations of the rational interfaces of the type I twins and irrational interfaces of the type II twins are based on this hypothesis. Recently, some asymmetric facets and unconventional twins have been observed in magnesium that do not agree with this paradigm. Here, we show that these features can be explained by admitting that some of the facets of the interface or the interface itself may be a rational plane that can be slightly distorted and transformed into a new rational plane with non-equivalent Miller indices. Relaxing the invariant plane assumption permits to broaden the usual lattice theory of twinning and include these unconventional "weak" interfaces and twins. A computer program was written to calculate the type I, type II and weak twins. It explains the asymmetric facets, the unconventional deformation twins, and the "deformation graining" phenomenon recently reported in magnesium. (c) 2022 The Author(s). Published by Elsevier Ltd on behalf of Acta Materialia Inc. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )