Former mechanic here. There are two pieces to this:
1. Why do we push instead of pull: we push the lever to go to a larger sprocket because the chain needs that force to leap to a larger sprocket. Going to a smaller sprocket doesn’t require a much force because gravity takes over. All you have to do to go to a smaller sprocket is move the chain and it will naturally go down but it needs more push to go up and our thumbs are better at pushing.
2. Why does this have the opposite effect for front/back left/right: in the front (left) the bigger gear pulls more chain per pedal revolution. The back (right) the bigger gear takes more chain to complete a revolution of the wheel. If the front is pulling more chain links and the back takes less links to rotate the wheel, you go fast. If the font is pulling less chain and the rear takes more links to rotate the wheel, you goo slow.
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