Given a finite CW or simplicial decomposition of a space $X$ and a ring homomorphism $\varphi:\mathbb{Z}[\pi_1(X)]\to F$ for a field $F$, if the $\varphi$-twisted homology is trivial, then the Reidemeister-Franz torsion $\tau^\varphi(X)\in F$ is an invariant of the twisted chain complex, well-defined up to multiplication by $\pm\varphi(\pi_1(X))$. (The value can be pinned down further by choosing a "homology orientation" and an "Euler structure".)

The definition of the torsion is rather opaque to me; it involves products of determinants of basis-change matrices. I vaguely see that it has something to do with how well a splitting of the acyclic chain complex respects the cellular basis.

My question is whether there is a geometric interpretation of Reidemeister-Franz torsion. For instance, what is it measuring about the space? Is there an object in $X$ representing the torsion? Feel free to restrict the category if it helps.