We investigate waves close to the lower-hybrid frequency in 12 magnetotail reconnection electron diffusion region (EDR) events with guide field levels of near-zero to 30%. In about half of the events, the wave vector has a small component along the current sheet normal, consistent with known lower-hybrid drift wave properties, but the perpendicular magnetic field fluctuations can be comparable or greater than the parallel component, a feature unique to the waves inside and adjacent to EDRs. Another new wave property is that the wave vector has a significant component along the current sheet normal in some events and completely along the normal for one event. In 1/4 of the events, the 𝐴𝐴∇⋅𝑷𝑷𝑒𝑒 term has a significant contribution to the wave electric field, possibly a feature of lower-hybrid waves more likely to exist in the diffusion region than further away from the X-line. Electron temperature variations are correlated with the wave potential, due to wave electric field acceleration and crossings at the corrugated separatrix region with different amounts of mixing between reconnection inflowing and outflowing populations. The latter also leads to the anti-correlation between parallel and perpendicular temperature components. Using four-spacecraft measurements, the magnetic field line twisting is demonstrated by the correlated fluctuations in 𝐴𝐴(∇×𝑽𝑽𝐸𝐸×𝐵𝐵)|| and 𝐴𝐴(∇×𝐁𝐁)||. The lower-hybrid wave in the EDR of weak guide field reconnection may be generated near separatrices and penetrate to the mid-plane or locally generated, and the latter possibility is beyond the prediction of previous reconnection simulations.