  We exploit very deep mid-IR (MIR) and X-ray observations by Spitzer and
Chandra in the GOODS North to identify signs of hidden (either starburst or
AGN) activity in spheroidal galaxies between z~0.3 and 1. Our reference is a
complete sample of 168 morphologically classified spheroidal galaxies with
z[AB]<22.5 selected from GOODS ACS imaging. Nineteen of these have 24 micron
detections in the GOODS catalogue, half of which have an X-ray counterpart in
the 2 Ms Chandra catalogue, while about 25% have 1.4 GHz fluxes larger than 40
microJy. Traces of hidden activity in the spheroidal population are also
searched for in the deep X-ray images and 14 additional galaxies are detected
in X-rays only. The nature of the observed MIR emissions is investigated by
modelling their SEDs based on the available multi-wavelength photometry,
including X-ray, UV, optical, near-IR, MIR and radio fluxes, and optical
spectroscopy. The amount of dust derived from the IR emission observed by
Spitzer appears in excess of that expected by mass loss from evolved stars. Our
analysis of the X-ray and MIR properties leads us to conclude that at least 8
of the 19 24-um bright sources should hide an obscured AGN, while the X-ray
undetected sources are more likely dominated by star formation. [abridged] We
conclude that ~30 objects (~20%) of the original flux-limited sample of 168
spheroidal galaxies in the GOODS-North are detected during phases of prominent
activity, of both stellar and quasar origin. Due to the short expected
lifetimes of the IR and X-ray emissions, this fraction might imply a
significant level of activity in this class of galaxies during the relatively
recent cosmic epochs, z~0.3 to z~1, under investigation.
