Comet 8P/Tuttle is a nearly isotropic comet whose physical properties are poorly known and might be different from those of ecliptic comets owing to their different origin. Two independent observations have shown that 8P/Tuttle has a bilobate nucleus. Our goal is to determine the physical properties of the nucleus (size, shape, thermal inertia, and albedo) and coma (water and dust) of 8P/Tuttle. We observed the inner coma of 8P/Tuttle with the infrared spectrograph and the infrared camera of the Spitzer Space Telescope. We obtained one spectrum (5–40 μm) on 2 November 2007 and a set of 19 images at 24 μm on 22–23 June 2008 sampling the rotational period of the nucleus. The data were interpreted using thermal models for the nucleus and the dust coma, and we considered two possible shape models of the nucleus derived from Hubble Space Telescope visible and Arecibo radar observations. We favor a model for the nucleus shape that is composed of two contact spheres with respective radii of 2.7 ± 0.1 km and 1.1 ± 0.1 km and a pole orientation with RA = 285 ± 12° and Dec = +20 ± 5°. The thermal inertia of the nucleus lies in the range 0–100 J K−1 m−2 s−1∕2 and the R-band geometric albedo is 0.042 ± 0.008. The water production rate amounts to 1.1 ± 0.2 × 1028 molecules s−1 at 1.6 AU from the Sun pre-perihelion, which corresponds to an active fraction of ≈9%. At the same distance, the ɛfρ quantity amounts to 310 ± 34 cm, and it reaches 325 ± 36 cm at 2.2 AU post-perihelion. The dust grain temperature is estimated to be 258 ± 10 K, which is 37 K higher than the thermal equilibrium temperature at 1.6 AU. This indicates that the dust grains that contribute to the thermal infrared flux have a typical size of ≈10 μm. The dust spectrum exhibits broad emission around 10 μm (1.5σ confidence level) and 18 μm (5σ confidence level) that we attribute to amorphous pyroxene.