Hi,
From what you say you are probably playing four-track stereo tapes with a two-track playback head.
A two-track playback head with 2mm guard band between tracks in the middle of the tape (standard two-track format a la Tape Project, as opposed to wide-track two track which only has a 0.75mm guard band) will cover track 1 and track 4 of a four-track tape, with tracks 2 and 3 mostly falling in the guard band.
Since four-track stereo tapes are recorded with left and right channels on tracks 1 and 3 "going" and tracks 4 and 2 "coming" (accessible by flipping the reels over on a two-channel machine), you are hearing the left channel of the "going" stereo programme on track 1 in your left channel, and the left channel of the "coming" stereo programme on track 4 backwards in your right channel.
You need to find a machine with a four-track stereo playback head. Consumer machines, as opposed to a pro machine like the Otari, are your best bet, since four-track stereo was the standard consumer format, with two-track more of a semi-pro or pro format.
Revox, for example, made machines in both two- and four- track formats; you need to find one in the format you want. Some machines by other manufacturers had two playback heads, selectable by switching.
Best regards,