  We report on the observation of terahertz radiation induced photoconductivity
and of terahertz analog of the microwave-induced resistance oscillations (MIRO)
in HgTe-based quantum well (QW) structures of different width. The MIRO-like
effect has been detected in QWs of 20 nm thickness with inverted band structure
and a rather low mobility of about 3 $\times$ 10$^5$ cm$^2$/V s. In a number of
other structures with QW widths ranging from 5 to 20 nm and lower mobility we
observed an unconventional non-oscillatory photoconductivity signal which
changes its sign upon magnetic field increase. This effect was observed in
structures characterized by both normal and inverted band ordering, as well as
in QWs with critical thickness and linear dispersion. In samples having Hall
bar and Corbino geometries an increase of the magnetic field resulted in a
single and double change of the sign of the photoresponse, respectively. We
show that within the bolometric mechanism of the photoresponse these unusual
features imply a non-monotonic behavior of the transport scattering rate, which
should decrease (increase) with temperature for magnetic fields below (above)
the certain value. This behavior is found to be consistent with the results of
dark transport measurements of magnetoresistivity at different sample
temperatures. Our experiments demonstrate that photoconductivity is a very
sensitive probe of the temperature variations of the transport characteristics,
even those that are hardly detectable using standard transport measurements.
