  We examine the dynamics of a closed cosmology whose matter source is that of
a conformally coupled scalar field with a broken ${\rm SO}(1,1)$ symmetry,
which correspond to the $\alpha$-attractors proposed by Linde and Kallosh.
Following a field redefinition, such models give rise to "T-model" inflationary
potentials, whose dynamics provide both an inflationary phase and a classical
bounce. We show that the universe can undergo bounces far from the regime of
quantum gravity. We analyse perturbations on this background with particular
attention given to the effects of a double-bouncing scenario (with rapid
recollapse between bounces) on the long wavelength modes. We demonstrate that
the predictions of such models agree well with observations and might explain
the suppression of power in the low multiples of the CMB.
