The exact construction is the expression "manquer à + quelque chose/quelqu’un".
Tu manques à moi
(litterally in English: You are missing to me)
is not used that way. However we see that "tu" is subject and "moi" is a "complément d’objet indirect" which can be put in front of the verb, as shown below :
Tu me manques
I think the problem comes from how French and English approach the verb and which word is the subject.
Edit: As seen on this page and to help in the understanding, in English to miss can be read as "fail to make contact with", or in our case "be sorry to be without". The French verb "manquer" is to be read as "to be lacking/absent". "Tu me manques" meaning "You are lacking in my life", you are the person "performing" the action of ‘being absent in my life’ right now.
I do know you / "Je connais toi" / Je te connais
I do see you / "Je vois toi" / Je te vois
(Could we say in English something like : I do do my homework? seems weird :P)
But in French, the subject is the person "verb-ing" if I may say.
Who is seeing you: I. Who is knowing you : I.
In "I miss you" : Who is missing : you.
The subject "is missing", is not here. And he’s missing to someone.
To solve your corollary, you should say "Je te manque" which is saying "Je manque à toi" (I am missing to you)
I do not have any other verb in mind. The explanation is not backed up by proper grammatical references, but I hope it will help you get a clearer picture of the situation.