It is well known that symmetry considerations can lead to improved bounds on, or even determine, the conductivity of two-component symmetric materials. The present work exploits symmetry properties to derive explicit higher-order bounds for three-component symmetric materials. The bounds contain geometric parameters. But even without any knowledge of these geometric parameters, substantial improvement on previous bounds is made. This is discussed in the context of equiaxed polycrystals. Results include a parameter-independent pair of bounds that for some polycrystals becomes third-order, and a parameter-dependent third-order upper bound that can be partially attained.