We report on a compact sensor by integrating a Mach-Zehnder interference and a cladding Bragg grating in a same section of all-solid photonic bandgap fiber. Theoretical investigation reveals that the Bragg grating resonance stems from the coupling of counter-propagating cladding LP01-like supermodes and the Mach-Zehnder interference works between a LP01-like supermode and LP01 core mode. Compared with the interference fringe, such supermode grating dip responses to axial strain in a more sensitive and opposite-direction manner. Whereas, the interference fringe shows a higher temperature sensitivity than the supermode grating dip. By means of these different responses, this device finds a useful application in the discrimination of temperature and axial strain.