Abstract: In this talk, I will overview my recent works on up-down quark matter and the associated new possible types of compact stars, such as up-down quark stars and the inverted hybrid stars that we termed ``cross stars” (CrSs). CrSs consist of a hadronic matter core and a quark matter crust, with an inverted structure compared to conventional hybrid stars. This distinct stellar structure naturally arises from the quark matter to hadronic matter transition associated with the chemical potential crossing, in the context of stable quark matter hypothesis. We find that the interplay between the hadronic matter and quark matter compositions of CrSs can help to reconcile various recent astrophysical constraints. This leaves more space open for the equations of states of both hadronic matter and quark matter. Finally, I will present some of my recent results on the radial oscillations of CrSs.