The first part of this paper contains new mathematical techniques for describing a spacetime anisotropy as suggested by the violation of parity conservation. Geometric measures of spacetime involve both the laboratory doing them and the events upon which these measures are done. The time form c and the spacelike length γ are the basic issues of those measures. Both depend on events and also on the timelike direction of the laboratory. Relativity tells that the field γ−c ⊗c depends, on the contrary, on events only; in this sense, relativistic spacetime is isotropic. If γ and c do not have that property, the manifold where the observable geometry takes place must be the set of timelike directions. The geometric structure of this manifold given by c and γ is studied in detail. The second part of the paper contains the study of a line of thought opposite to chronogeometry: Building the geometry from lengths instead of times. The datum is γ; through the conditions of stationary spacelike volume and of stationary proper time, a class of time forms and a gauge are obtained under some weak restrictions. Newtonian and relativistic spacelike metrics fulfill these restrictions. Standard connections are induced; they define the absolute derivative of physical fields and the geometric structure of the manifold of timelike directions. The paper ends with some comments about the remaining problem: to suggest and justify field equations.