Tunability is essential for unlocking a range of practical applications of high-efficiency metasurface-based nanophotonic devices and systems. Increased research efforts in this area during recent years led to significant progress regarding tuning mechanisms, speed, and diverse active functionalities. However, so far almost all the demonstrated works are based on a single type of physical stimulus, thereby excluding important opportunities to enhance the modulation range of the metadevices, the available design options, as well as interaction channels between the metadevices and their environment. In this article, it is experimentally demonstrated that multi-responsive metasurfaces can be realized by combining asymmetric, highly resonant metasurfaces with multi-responsive polymeric materials. The respective copolymers combine light- and temperature-responsive comonomers in an optimized ratio. This work demonstrates clearly reversible light-responsive, temperature-responsive, and co-responsive tuning of the metasurface optical resonance positions at near-infrared wavelengths, featuring maximum spectral resonance shifts of nearly twice the full-width-at-half-maximum and accompanied by more than 60% absolute modulation in transmittance. This work provides new design freedom for multifunctional metadevices and can potentially be expanded to other types of copolymers as well. Furthermore, the studied hybrid multiresponsive systems are promising candidates for multi-dimensional sensing applications.