Data Collaboration (DC) enables multiple parties to jointly train a model without exposing their private datasets. Each party privately transforms its data using a secret linear basis and shares only the resulting intermediate representations. Existing theory asserts that any target basis spanning the same subspace as the secret bases should suffice; however, empirical evidence reveals that the particular choice of target basis significantly influences model accuracy and stability. In this paper, we introduce Orthonormal Data Collaboration (ODC), a novel DC framework that explicitly enforces orthonormality constraints on both the secret and target bases. Under these constraints, the basis alignment step reduces precisely to the classical Orthogonal Procrustes Problem, admitting a closed-form solution. We rigorously establish that the resulting orthonormal change-of-basis matrices achieve orthogonal concordance, aligning all parties’ intermediate representations up to a common orthogonal transformation. Consequently, downstream model performance becomes invariant to the specific choice of orthonormal target basis. Computationally, ODC substantially reduces alignment complexity from O(\min\{a(cl)^2,a^2cl\}) to O(acl^2) where a denotes anchor data size, l the latent dimension, and c the number of collaborating parties. Extensive empirical evaluations confirm the theoretical advantages of ODC, demonstrating alignment speed-ups of up to two orders of magnitude compared to state-of-the-art DC methods, alongside comparable or superior accuracy across multiple benchmark datasets. ODC maintains robust privacy under the semi-honest threat model and requires only a single round of communication. These results establish ODC as a practically advantageous and computationally efficient enhancement to existing DC pipelines, particularly when orthonormal secret bases are naturally feasible.