Yes, there is: if you just say "Io leggo" it means merely that you read, nothing more. But if you say "leggo io" it means that you are telling someone else that you want to read, you are stressing that, maybe because you don't want someone else to read so you are telling someone that YOU want to read: "Guarda, tua moglie ha lasciato una lettera" "Aspetta, leggo io!" "Io chiamo Marco" "Marco lo chiamo io, perché è un mio amico!"

As usual reversing the order adds emphasis. Why is emphasis added depends upon the context, usually it is related to some kind of emotions related to the matter, like: -- forbidding the other person to do the thing -- the desire to do the thing (or the task) in order to give an help to the other person -- being the one more qualified to do the task. Andrea already gave very good examples of these situations.