In scenarios that stabilize the electroweak scale via either supersymmetry or Higgs compositeness, the top quark is typically accompanied by "top partner" particles that cancel the large quantum contributions of the top quark to the Higgs mass. Barring fine tuning, these particles are expected to have masses below a TeV and their signatures serve as an important target for ongoing LHC searches. While many different scenarios are being considered in terms of mass spectra and couplings, the top partner is always assumed to be either a color-triplet scalar with charge 2/3 or a color-triplet fermion with several specific possibilities for the charge. In this talk, I will discuss models, based on a larger symmetry structure than is usually assumed, in which either a scalar or a fermionic top partner can have any arbitrary value for the electric charge. The phenomenology of these models is in general quite different from the standard scenarios. While the very diverse arsenal of the existing LHC searches still provides significant coverage, some signatures, even very simple ones, are not being searched for efficiently. I will then focus on one such signature, pairs of top+jet resonances, and analyze the possibility of addressing this signature by differential measurements of ttbar samples.