© 2018 authors. Published by the American Physical Society. Crab crossing is essential for high-luminosity colliders. The high-luminosity Large Hadron Collider (HL-LHC) will equip one of its interaction points (IP1) with double-quarter wave (DQW) crab cavities. A DQW cavity is a new generation of deflecting rf cavities that stands out for its compactness and broad frequency separation between fundamental and first high-order modes. The deflecting kick is provided by its fundamental mode. Each HL-LHC DQW cavity shall provide a nominal deflecting voltage of 3.4 MV, although up to 5.0 MV may be required. A proof-of-principle (POP) DQW cavity was limited by quench at 4.6 MV. This paper describes a new, highly optimized cavity, designated the DQW SPS series, which satisfies dimensional, cryogenic, manufacturing, and impedance requirements for beam tests at the Super Proton Synchrotron (SPS) and operation in the LHC. Two prototypes of this DQW SPS series were fabricated by U.S. industry and cold tested after following a conventional superconducting radio-frequency surface treatment. Both units outperformed the POP cavity, reaching a deflecting voltage of 5.3-5.9 MV. This voltage - the highest reached by a DQW cavity - is well beyond the nominal voltage of 3.4 MV and may even operate at the ultimate voltage of 5.0 MV with a sufficient margin. This paper covers fabrication, surface preparation, and cryogenic rf test results and implications.