We study the special form that the general multi-image tensor formalism takes under the plane + parallax decomposition, including matching tensors and constraints, closure and depth recovery relations, and inter-tensor consistency constraints. Plane + parallax alignment greatly simplifies the algebra, and uncovers the underlying geometric content. We relate plane + parallax to the geometry of translating, calibrated cameras, and introduce a new parallax-factorizing projective reconstruction method based on this. Initial plane + parallax alignment reduces the problem to a single rank-one factorization of a matrix of rescaled parallaxes into a vector of projection centres and a vector of projective heights above the reference plane. The method extends to 3D lines represented by via-points and 3D planes represented by homographies.