Something I've noticed whilst I'm learning Italian is that when you have a phrase such as "ti chiamo" is that the action described is being applied to the object of the sentence, and the conjugation determines who is performing the action ("I call you" in this instance).
However, "mi piace" (or even "mi piacciono") seems to be an exception to this, as it means "I like". My question is whether this reversal is rare or I just haven't come across examples where this structure is used.
The peculiar structure with verb "piacere" you noticed is due to the fact that literal translation of this Italian verb is "to be pleasing to (someone)", so that this "someone" is the indirect object of the sentence and not the subject.
As the book Italian Verbs for Dummies explains (I have done a correction to the text because there was a confusion between the words "wine" and "milk")
As a verb, piacere (to like, to be pleasing to) is unique in that it takes indirect-object pronouns rather than personal pronouns. For example, "I like wine" becomes "Wine is pleasing to me," or "Mi piace il vino." Similarly, you need the preposition "a" (to) when using proper nouns. For example, you translate "A Domenico piace la pasta" as (literally) Pasta is pleasing to Domenick (or Domenick likes pasta).
As @DaG has pointed out in a comment, "personal pronouns" in the above extract really refers to "subject pronouns".
At this book you can also read
With the verb piacere, what's typically the object in an English construction becomes the subject in an Italian construction. For example, I like spaghetti becomes (literally) To me, spaghetti is pleasing, or "Mi piacciono gli spaghetti."
The same book also explains that an Italian verb which behaves in a similar way is "mancare", used to convey the meaning of "to miss" or "to lack":
The verb mancare, which means to miss or to lack, functions in precisely the same manner as the verb piacere (to like). In other words, you generally use it with indirect objects and indirect-object pronouns, and it translates literally as Something is missing to me.
An example of use of "mancare" which come from this book is the following:
Cara mamma e papa, mi mancate! (Dear mom and dad, I miss you!)
The literal translation of "mi mancate" would be "you (subject) are missing to me (indirect object)".
Other examples of "mancare" coming from the same book are:
Io ti manco? (Do you miss me? Literally: Am I missing to you?)
Sì, tu mi manchi. (Yes, I miss you.)
Mi manca mia madre. (I miss my mother.)
You are just used to the English construction, but I'm not sure if it is more natural than the other. Compare with the English sentence Basketball pleases/interests/attracts me. What is the action here? What should be the subject? Who or what "does" something? Who is the actor? Basketball, for attracting / interesting me, or me, the one who likes basketball?
That's just how these verbs are constructed in different languages; it's difficult to find a logic in it. Constructions similar to the Italian one appear in many other languages; for instance French (X me plait), Spanish (Me gusta X), and German (X gefällt mir), so it's not that crazy.
Another analogous sentence that is constructed differently in various languages is I'm hungry, which is ho fame in Italian (literally I have hunger), and similarly in French (j'ai faim) and German (Ich habe hunger), and Spanish (tengo hambre). Is hunger something you have (like in "I have no desire to eat"), or a state of being? Both viewpoints seem equally valid, according to people with different native languages.
Just to add some thought in simple words...
"Piacere" literally means "to please, to give pleasure". It is correct to say "La pasta piace" (it gives pleasure, said in an indefinite general form); but to say who it pleases, one must add an "a" followed by the entity the pasta is pleasing: "la pasta piace alla gente", or "piace a me". I think this construction expresses well the passivity of the person liking pasta: a person can do nothing about it, it is a victim of the simple fact the pasta is so pleasing (and often the victim pays this passivity with a raising weight!).
The english translation of "I like pasta" is "Io amo la pasta" (I love pasta) or, with less sentiment, "Io apprezzo la pasta" (I appreciate). "Amare" is a special verb, because is the only one able to express love (sentimental love) to another person, and probably this is the reason why, referring for example to pasta, an italian will likely choose to say "apprezzo" or "mi piace".
I find the same passivity in "mancare" (to miss). Again, if I like pasta but I can not have it, I am a victim, and the subject is the pasta ("a me manca la pasta"). Another example is "sembrare" (to seem, to look like) which, luckily, has the same construction in english too. Would an english speaker wonder about "sembrare"? Other verbs having similar construction, some found in other answers in this page, are apparire (to appear), attrarre (to attract), eccitare (to excite), trascinare (to drag).
