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We study the low-temperature heat transport in clean two-leg spin laddercompounds coupled to three-dimensional phonons. We argue that the very largeheat conductivities observed in such systems can be traced back to theexistence of approximate symmetries and corresponding weakly violatedconservation laws of the effective (gapful) low--energy model, namelypseudo-momenta. Depending on the ratios of spin gaps and Debye energy and onthe temperature, the magnetic contribution to the heat conductivity can bepositive or negative, and exhibit an activated or anti-activated behavior. Inmost regimes, the magnetic heat conductivity is dominated by the spin-phonondrag: the excitations of the two subsystems have almost the same driftvelocity, and this allows for an estimate of the ratio of the magnetic andphononic contributions to the heat conductivity.Comment: revised version, 8 pages, 3 figures, added appendi

Heat transport properties of clean spin ladders coupled to phonons: Umklapp scattering and drag