Orthogonal frequency-division multiplexing (OFDM) is a dominant waveform in modern wireless systems, yet its high peak-to-average power ratio (PAPR) and limited adaptability hinder efficient support for integrated communication and sensing. This paper proposes deep block-unitary precoded OFDM (DBU-OFDM), a structure-preserving learning framework that enables trainable waveform adaptation while preserving the DFT-based signal structure, pilot/null resource protection, and compatibility with low-complexity frequency-domain equalization. The proposed design restricts learning to a block-unitary transformation over data subcarriers and preserves pilot and null resources for structural compatibility. The transform is parameterized by recursive Householder reflections, ensuring strict unitarity as well as differentiable, numerically stable, and complexity-controllable implementation. Results show that DBU-OFDM achieves PAPR tails close to block-pilot DFT-s-OFDM while retaining comb-type pilots, improves communication reliability in frequency-selective fading via frequency-domain diversity, and enhances range and velocity estimation in direct sensing, especially in dimension-limited settings. Over-the-air USRP experiments and FPGA prototyping further verify its practical feasibility, demonstrating low error vector magnitude (EVM), clear PAPR reduction in real transmission, and hardware throughput up to 200~MS/s with microsecond-level latency. DBU-OFDM therefore offers a practical intermediate solution between conventional model-based OFDM waveforms and unconstrained neural transceivers for next-generation integrated communication and sensing systems.