Linear latent coordinates are not unique without constraints

In the observation model $y=Cz+b$, let $Q$ be an orthogonal matrix. Replacing $z$ by $Qz$ and $C$ by $C{Q}^T$ leaves $y$ unchanged because $Q^{T}Q=I$. Thus reconstruction alone does not give each latent axis a unique biological meaning. A particular stochastic model can restrict this symmetry through its prior and parameter constraints; the algebra does not claim every fitted latent model has every rotational ambiguity.