  Circinus X-1 is a neutron-star-accreting X-ray binary in a wide (P$_{\rm
orb}$ = 16.6 d), eccentric orbit. After two years of relatively low X-ray
luminosity, in May 2010 Circinus X-1 went into outburst, reaching 0.4 Crab
flux. This outburst lasted for about two orbital cycles and was followed by
another shorter and fainter outburst in June. We focus here on the broadband
X-ray spectral evolution of the source as it spans about three order of
magnitudes in flux. We attempt to relate luminosity, spectral shape, local
absorption, and orbital phase. We use multiple Rossi-XTE/PCA (3.0--25 keV) and
Swift/XRT (1.0--9.0 keV) observations and a 20 ks long Chandra/HETGS
observation (1.0--9.0 keV), to comprehensively track the spectral evolution of
the source during all the outbursting phases. These observations were taken
every two/three days and cover about four orbital cycles. The PCA data mostly
cover the major outburst, the XRT data monitor the declining phase of the major
outburst and all the phases of the minor outburst, and Chandra data provide an
essential snapshot of the end of this overall outbursting phase. The X-ray
spectrum can be satisfactorily described by a thermal Comptonization model with
variable neutral local absorption in all phases of the outburst. No other
additive component is statistically required. The first outburst decays
linearly, with an ankle in the light curve as the flux decreases below
$\sim$\,5 $\times$ 10$^{-10}$ erg cm$^{-2}$ s$^{-1}$. At the same time, the
source shows a clear spectral state transition from an optically thick to an
optically thin state. While the characteristics of the first, bright, outburst
can be interpreted within the disk-instability scenario, the following, minor,
outburst shows peculiarities that cannot be easily reconciled in this
framework.
