Among the theoretically predicted two-dimensional topological insulators, InAs/GaSb double quantum wells (DQWs) have a unique double-layered structure with electron and hole gases separated in two layers, which enables tuning of the band alignment via electric and magnetic fields. However, the rich trivial-topological phase diagram has yet to be experimentally explored. We present an in situ and continuous tuning between the trivial and topological insulating phases in InAs/GaSb DQWs through electrical dual gating. Furthermore, we show that an in-plane magnetic field shifts the electron and hole bands relatively to each other in momentum space, functioning as a powerful tool to discriminate between the topologically distinct states.