We developed CdTe double-sided strip detectors (DSDs or cross strip detectors) and evaluated their spectral and imaging performance for hard X-rays and gamma-rays. Though the double-sided strip configuration is suitable for imagers with a fine position resolution and a large detection area, CdTe diode DSDs with indium (In) anodes have yet to be realized due to the difficulty posed by the segmented In anodes. CdTe diode devices with aluminum (Al) anodes were recently established, followed by a CdTe device in which the Al anodes could be segmented into strips. We developed CdTe double-sided strip devices having Pt cathode strips and Al anode strips, and assembled prototype CdTe DSDs. These prototypes have a strip pitch of 400 micrometer. Signals from the strips are processed with analog ASICs (application specific integrated circuits). We have successfully performed gamma-ray imaging spectroscopy with a position resolution of 400 micrometer. Energy resolution of 1.8 keV (FWHM: full width at half maximum) was obtained at 59.54 keV. Moreover, the possibility of improved spectral performance by utilizing the energy information of both side strips was demonstrated. We designed and fabricated a new analog ASIC, VA32TA6, for the readout of semiconductor detectors, which is also suitable for DSDs. A new feature of the ASIC is its internal ADC function. We confirmed this function and good noise performance that reaches an equivalent noise charge of 110 e- under the condition of 3-4 pF input capacitance.