Don't apologise when you did wrong, apologise when someone thinks you did wrong.

Blessed is the servant who is not quick to excuse himself and who humbly accepts shame and blame for a sin, even though he did not commit any fault. —Francis of Assisi

When I realised the effect that has, I was astonished. When someone thinks you did something wrong, apologise, even if you didn't. Say, "I'm sorry." Say, "I'll try to do better." Say, "How can I fix it?" It turns out, it doesn't matter if you did wrong or not, it only matters that they think you did. No, it's not fair. Suck it up.

If you don't apologise now, and they later realise it wasn't your fault, they'll feel guilty, but it doesn't repair the initial damage. Contrariwise, if you sincerely apologise immediately, and do whatever you can, as though it were your fault, it makes them like you more immediately, and even more when they realise it wasn't your fault at all, and you still tried to help.

This has the side-effect of making you apologise for things that are your fault that you would've normally made excuses for.

Realising and internalising that seriously changed my relationship, with just about everyone, for the better.