We study the low-temperature heat transport in clean two-leg spin-ladder compounds coupled to threedimensional phonons. We argue that the very large heat conductivities observed in such systems can be traced back to the existence of approximate symmetries and corresponding weakly violated conservation laws of the effective gapful low-energy model, namely, pseudomomenta. Depending on the ratios of spin gaps and Debye energy and on the temperature, the magnetic contribution to the heat conductivity mag can be positive or negative and can exhibit an activated or antiactivated behavior. In most regimes, mag is dominated by the spin-phonon drag: the excitations of the two subsystems have almost the same drift velocity, and this allows for an estimate of the ratio mag/ph of the magnetic and phononic contributions to the heat conductivity.