In this paper, a new multiband fractal frequency selective surface (MF-FSS) is proposed. The work presents a new fractal design methodology for FSSs with Swastika fractal patch elements. The proposed MF-FSS includes periodic arrays of metallic patches, printed on a single layer substrate. The structure parametric analysis is carried out in terms of fractal iterations, geometry elements, and unit-cell size. The simple controllable feature of the proposed structure lets us tune the given parameters of FSS geometry to achieve adjustable bandstop filter. The fractal geometry allows us to design compact structures (CP = 54%) that behave like dual-polarized bandstop filters. The designed structure is validated by means of an excellent agreement between the simulation and measurement results. Also, results show the proposed structure presents the most desirable features like fractal compactness, multiband response, dual polarization, excellent angular stability, and controllability. This work presents a fundamental structure that can be applied into the more complex and sophisticated designs in future.