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Is that normal that only 1 side is responsible for a shared thing?
If they own it, yes.
Maybe there’s some weird system where they pay for it but everyone else owns it. Mmmm.
I’d want to know exactly what the situation is because if you have no real access to it nor influence over how it’s used you might find maintaining the wall of house difficult and if you wanted external insulation (you do) you couldn’t fit it.
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What's shown on the title plan/deeds, and is there any mention of a right of way over it? Might be worth purchasing the title plan/deeds for next door to check if there is any mention it there as well.
Hard to tell but it looks like both properties have gates to their rear gardens at the end of the passageway, so I would assume that the central of the fence runs through the middle of the passageway forming the boundary line.
Seems like the neighbour is pretty relaxed about bins being placed on their side but that could easily change should they sell etc.
Explicitly ask your solicitor. Assume nothing. You are paying them to answer your questions as much as get the legals right (and they are human [probably] so mistakes happen)
Nested parenthesis, what a mess.