We introduce STORMED hybrid games (SHG), a generalization of STORMED Hybrid Systems [21], which have natural specifications, allow rich continuous dynamics and admit various properties to be decidable. We solve the control problem for SHG using a reduction to bisimulation on game graphs. This reduction generalizes to a greater family of games, which includes o-minimal hybrid games [5]. We also solve the optimal-cost reachability problem for Weighted SHG and prove decidability of WCTL for Weighted STORMED hybrid systems.