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We analyze autocompensation possibilities in entanglement-based QKD protocols. In particular, we study the seminal BBM92 protocol and find that an autocompensating technique is possible, although with severe limitations. This prompts the introduction of a different, more practical protocol based on Bell state exchange parity (BSEP), which allows for intrinsic autocompensation of optical fiber perturbations in various two-dimensional fiber-optic encodings while retaining advantageous MDI-QKD characteristics. We present the BSEP protocol in detail, describing both the quantum light propagation and the optical hardware requirements. Finally, we analyze its security, computing its expected performance through the key rate.