Quantum computing platforms are susceptible to quantum-specific bugs (e.g., incorrect ordering of qubits or incorrect implementation of quantum abstractions), which are difficult to detect and require specialized expertise. The field faces challenges due to a fragmented landscape of platforms and rapid development cycles that often prioritize features over the development of robust platform testing frameworks, severely hindering the reliability of quantum software. To address these challenges, we present QITE, the first cross-platform testing framework for quantum computing platforms, which leverages QASM, an assembly-level representation, to ensure consistency across different platforms. QITE introduces the novel ITE process to generate equivalent quantum programs by iteratively (I)mporting assembly into platform representations, (T)ransforming via platform optimization and gate conversion, and (E)xporting back to assembly. It uses a crash oracle to detect failures during cross-platform transformations and an equivalence oracle to validate the semantic consistency of the final sets of assembly programs, which are expected to be equivalent by construction. We evaluate QITE on four widely-used quantum computing platforms: Qiskit, PennyLane, Pytket, and BQSKit, revealing 17 bugs, 14 of which are already confirmed or even fixed. Our results demonstrate QITE's effectiveness, its complementarity to existing quantum fuzzers in terms of code coverage, and its ability to expose bugs that have been out of reach for existing testing techniques.