  We propose a new stellar structure of compact stars, the ``cross stars" that
consist of a hadronic matter core and a quark matter crust, with an inverted
structure compared to the 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 the quark
matter hypothesis that either strange or up-down quark matter is the ground
state of baryonic matter at low pressure. We find that the interplay between
the hadronic matter and quark matter compositions of cross stars can help to
reconcile the small radii constraints indicated by the LIGO/Virgo GW170817
event, the large radii constraints set for massive compact stars by recent
NICER X-ray observations, and the recent observation of the most-massive pulsar
PSR J0952-0607. This leaves more space open for the equation of states of both
hadronic matter and quark matter.
