Tail-biting trellises are important graphical representations of codes, which can be simpler to decode than conventional trellises. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. In this paper we first develop a new algebraic framework for a systematic analysis of linear trellises which enables us to address open foundational questions. In particular, we present a useful and powerful characterization of linear trellis isomorphy. We also obtain a new proof of the linear factorization theorem of Koetter/Vardy, and point out unnoticed problems for the group case. Next, we apply our framework to: 1) describe all the factorizations of linear trellises; 2) determine and classify all the minimal linear trellises for a given linear code; and 3) prove that nonmergeable, one-to-one, linear trellises are determined by the edge-label sequences of certain closed paths. We also provide new insight into mergeability and state how results on reduced linear trellises can be extended to nonreduced ones. Our classification results are relevant to iterative/linear programming decoding, as we show that minimal linear trellises can yield different pseudocodewords even if they have the same graph structure.