  We develop theoretical aspects of refined Donaldson-Thomas theory for
threefold flops, and use these to determine all DT invariants for a doubly
infinite family of length 2 flopping contractions. Our results show that a
refined version of the strong-rationality conjecture of Pandharipande-Thomas
holds in this setting, and also that refined DT invariants do not classify
flops. Our main innovation is the application of tilting theory to better
understand the stability conditions and cyclic A-infinity-deformation theory of
these spaces. Where possible we work in the motivic setting, but we also
compute intermediary refinements, such as mixed Hodge structures.
