Phase retrieval is a prevalent problem in digital signal processing and experimental physics that consists of estimating a complex signal from magnitude measurements. This paper expands the classical phase retrieval framework to electric power systems with unknown network models and limited access to observations of voltage magnitudes, active power injections, and reactive power injections. The proposed method recovers the phase angles and the power-phase angle submatrices of the AC power flow Jacobian matrix. This is made possible by deriving topology and parameter-free expressions for the structural symmetries of the power flow Jacobian that do not depend on the phase angles. These physical laws provide structural constraints for the proposed phase retrieval method. The paper then presents sufficient conditions for guaranteed recovery of the voltage phase angles, which also depend solely on voltage magnitudes, active power injections, and reactive power injections. The method offers two significant benefits: both estimating the voltage phase angles and recovering the power flow Jacobian matrix, a basis for approximating the power flow equations. Simulations on widely studied open-source test networks validate the findings.