As galaxy surveys begin to measure the imprint of baryonic acoustic oscillations (BAO) on large-scale structure at the sub-percent level, reconstruction techniques that reduce the contamination from nonlinear clustering become increasingly important. Inverting the nonlinear continuity equation, we propose an Eulerian growth-shift reconstruction algorithm that does not require the displacement of any objects, which is needed for the standard Lagrangian BAO reconstruction algorithm. In our simulations, the algorithm yields 95% of the BAO signal-to-noise obtained from standard reconstruction. The reconstructed power spectrum is obtained by adding specific simple 3- and 4-point statistics to the pre-reconstruction power spectrum, making it very transparent how additional BAO information from higher-point statistics is included in the power spectrum through the reconstruction process. Analytical models of the reconstructed density for the two algorithms agree at second order. Based on similar modeling efforts, we introduce four additional reconstruction algorithms and discuss their performance.