Reciprocal and mutual I would say are mostly interchangeable words. In some rare cases mutual might mean both at the same time, and reciprocal might mean a favor done in return. I hope this helps! よろしくおねがいします！

The issue here is 'collocation': some words are often used together and some words are almost never used together. For example, English speakers often say "best friend", but do not often say "favourite friend". "[M]utual help" is a collocation; these words are often used together. "[R]eciprocal help" is not a common collocation. The words are not interchangeable in your sentence, though everyone would understand the meaning. "Reciprocal" and "trade" are often used together. At least in the UK, "mutual" has a special legal meaning when it is used with "trade" (perhaps similar to 相互 in Japanese 相互会社). So I think business experts would try not to use "mutual" interchangeably with "reciprocal". In less formal writing and speaking, "mutual" might be acceptable in this sentence. And I think that mutual is almost always written before a noun, except in a fixed expression ("the feeling is mutual"). This is not mentioned in the Oxford Advanced Learners Dictionary though, so I may be wrong on this point. I hope that helps!

I believe there is a difference that you should consider. The word reciprocal means something that is MATCHING, coordinate or duplicated. The word is used when talking about actions such as deals, deposits, visits, enforcement and trade. The word mutual means something that is SHARED. It is used for relationships between two people or for the feelings and views they hold in common. People can feel mutual distrust, mutual esteem, mutual dislike or mutual affection. I hope this helps