Oh dear. You've learned the importance of having everything in writing when it comes to agreements and getting sign off.
Especially when it comes to your neighbours.
The second biggest mistake is falling out with your neighbours.
When you agreed "like for like" - I'm assuming the better side of the fence was on their side?
When you agreed "like for like" - You forgot to mention you'd be installing concrete instead of wooden fence posts - you know the environmentally toxic substance that also not so great for humans.
If you agreed "like for like", you need to switch the panels around. Then I'd probably talk to the landscape gardener and grow some pretty vine plants up your side of the fence.
The third problem you've not yet forseen is who maintains the fence from this point forward? I'd get that in writing before you switch any fence panels around.
We also paid 50% to replace our neighbour's back fence. Mainly because our dogs (ours much larger) kept jumping up to see each other and damaging the fence and flowerbeds. We thought a slightly higher fence would solve the problem. It did. But we got a fencing contractor quote. We agreed on fence panels and posts and price and we also put it in writing, that as much as we're financially contributing to the fence, we accept no responsibility for its onwards maintenance.
Next time state your wants clearly - "We want a more attractive fence boundary for our garden. This is what we propose. This is the quote from a professional contractor for a professional job. However, we want the best side of the fence to face us. If you agree, we'll pay for the entire fence, and install concrete posts, which makes it far cheaper to maintain/replace the fence, but moving forwards, you'll be responsible for the maintainance of the fence."