Lemma 3.1. Take a two-taxon discrete-time Markov model that is defined by a root distribution π m = ( π $^{m}$(1) . . . π $^{m}$(4) ) and two 4 × 4 probability transition matrices P ma and P $^{mb}$. Define Π m = diag π $^{m}$. Assume Then π $^{m}$, P $^{ma}$, and P mb are uniquely determined by the joint probability matrix J ab = ( P mb ) $^{⊺}$Π $^{m}$Pma up to one of eight permutations of the states at node m . That is, the model is identifiable up to a suitable reordering of internal states. (a) P ma and P mb are strand-symmetric; (b) π m is compositionally asymmetric; and (c) P ma and P mb are invertible. Example 3.1. It is interesting that a non-reversible, stationary, strand-symmetric model is not identifiable for two taxa, as can be shown by counterexample. Take such a model defined by and π m = ( 0 . 20 0 . 30 0 . 30 0 . 20 ) P ma = P mb =    $^{0}$$^{.}$$^{38}$ 0 . 25 0 . 21 0 . 16 0 . 16 0 . 43 0 . 27 0 . 14 0 . 14 0 . 27 0 . 43 0 . 16 0 . 16 0 . 21 0 . 25 0 . 38     that yields the joint probability distribution J ab = ( P mb ) $^{⊺}$Π P $^{ma}$. The model defined 9