  We present statistics of HeII Lya transmission spikes and large-scale
absorption troughs using archival high-resolution ($R=\lambda /\Delta \lambda
\simeq 12,500$-$18,000$) far-UV spectra of eight HeII-transparent quasars
obtained with the Cosmic Origins Spectrograph on the Hubble Space Telescope.
The sample covers the redshift range 2.5<z<3.8, thereby probing the rapidly
evolving HeII absorption at the end of the HeII reionization epoch. The
measured lengths of the troughs decrease dramatically from L>100cMpc at z>3 to
L~30cMpc at z~2.7, signaling a significant progression of HeII reionization at
these redshifts. Furthermore, unexpectedly long L~65cMpc troughs detected at
z~2.9 suggest that the UV background fluctuates at larger scales than predicted
by current models. By comparing the measured incidence of transmission spikes
to predictions from forward-modeled mock spectra created from the outputs of a
(146cMpc)^3 optically thin Nyx hydrodynamical simulation employing different UV
background models, we infer the redshift evolution of the HeII photoionization
rate $\Gamma_\mathrm{He\,II}(z)$. The photoionization rate decreases with
increasing redshift from $\simeq 4.6\times 10^{-15}\mathrm{\,s^{-1}}$ at z~2.6
to $\simeq 1.2 \times 10^{-15}\mathrm{\,s^{-1}}$ at z~3.2, in agreement with
previous inferences from the HeII effective optical depth, and following
expected trends of current models of a fluctuating HeII-ionizing background.
