This paper investigates the constructions of tail-biting trellises for linear block codes as introduced by Koetter and Vardy (2003) and Nori and Shankar (2006). For a given code, the sets of characteristic generators are defined slightly more generally than by Koetter and Vardy. In particular, they are not uniquely determined by the code. The effect of the choice of characteristic generators on the resulting product trellises, called KV-trellises, is discussed in detail. It is shown that each KV-trellis is a span-based BCJR-trellis and that the latter are always nonmergeable. Finally, a duality conjecture posed by Koetter and Vardy is addressed by making use of a dualization technique of BCJR-trellises. The conjecture is proven for minimal trellises.