The two maps share the same center, resolution, rotation and layers.
See the side-by-side.js source to see how this is done.
Shift+double-click, mouse wheel, +/- keys; Shift+dragAlt+Shift+drag, r to resetb/B/c/C keys (WebGL only)h/H/s/S keys (WebGL only)o/O keysv/V keysj/l/m/x/L/M/X keys0 key