  We examine the quantum dynamics of cold atoms subjected to {\em pairs} of
closely spaced $\delta$-kicks from standing waves of light, and find behaviour
quite unlike the well-studied quantum kicked rotor (QKR). Recent experiments
[Jones et al, {\em Phys. Rev. Lett. {\bf 93}, 223002 (2004)}] identified a
regime of chaotic, anomalous classical diffusion. We show that the
corresponding quantum phase-space has a cellular structure, arising from a
unitary matrix with oscillating band-width. The corresponding eigenstates are
exponentially localized, but scale with a fractional power, $L \sim
\hbar^{-0.75}$, in contrast to the QKR for which $L \sim \hbar^{-1}$. The
effect of inter-cell (and intra-cell) transport is investigated by studying the
spectral fluctuations with both periodic as well as `open' boundary conditions.
