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We investigate the generation and temperature dependent evolution of spin photocurrents in WTe₂ with the circular photogalvanic effect. A large transverse spin photocurrent is observed when the light scattering plane coincides with one of the high-symmetry planes of the crystal, consistent with symmetry arguments. Upon lowering the temperature, its magnitude starts to decrease below 150 K, which coincides with the temperature below which hole pockets start to form in the Fermi surface. Around 50 K, the spin photocurrent switches its polarity and flows in the opposite direction. A similar behavior is observed in MoTe₂.