We propose a method for separating and detecting the nontensor modes of stochastic gravitational-wave backgrounds (SGWBs) using networks of space-based gravitational-wave detectors. We consider four distinct data-reconstruction schemes for the coinclination and anti-inclination orbital configurations of the LISA-Taiji network. We find that the coinclination configuration offers its advantages over the anti-inclination one and can achieve signal-to-noise ratios up to 17.3 for the vector modes and 10.4 for the scalar modes with the energy density spectrum as ${\mathrm{\Omega }}_{\mathrm{GW}}^p(f)={10}^{-12}$. Our method can be used to measure beyond-general-relativity polarization modes of SGWBs at the mHz frequency band, opening a new avenue for testing alternative gravity theories.

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• Cosmology
• General relativity
• Gravitational wave detection