Variable-speed-of-light (VSL) cosmologies are currently attracting interest as an alternative to inflation. We investigate the fundamental geometrodynamic aspects of VSL cosmologies and provide several implementations which do not explicitly break Lorentz invariance (no ``hard'' breaking). These ``soft'' implementations of Lorentz symmetry breaking provide particularly clean answers to the question ``VSL with respect to what?.'' The class of VSL cosmologies we consider are compatible with both classical Einstein gravity and low-energy particle physics. These models solve the ``kinematic'' puzzles of cosmology as well as inflation does, but cannot by themselves solve the flatness problem, since in their purest form no violation of the strong energy condition occurs. We also consider a heterotic model (VSL plus inflation) which provides a number of observational implications for the low-redshift universe if χ contributes to the ``dark energy'' either as CDM or quintessence. These implications include modified gravitational lensing, birefringence, variation of fundamental constants and rotation of the plane of polarization of light from distant sources.