In this work, the electromagnetic response of a mathematically interesting shape-a Möbius strip-is presented, along with a ring resonator for comparison. Both resonators consist of a central lossy dielectric layer bounded by perfectly conducting layers. For the case of the Möbius strips, the computational results show that there are a family of half-integer wavelength modes within the dielectric layer. These additional modes result in increased absorption, and a corresponding reduction in the radar cross section. Interestingly, rotational scans show that these modes can be excited over a large angular range. This investigation gives an understanding of the electromagnetic response of these structures, paving the way for future experiments on Möbius strip resonators.