It took a very long time for writing to become standardized in the Greek world. Some people wrote left-to-right, others right-to-left and many changed between the two at the end of each line (this is called boustrophedon.

When they changed the direction of the writing they also changed the direction of the letters. Thus you will see items in boustrophedon that have mirrored letters.

One of the oldest examples of Greek writing actually comes from the island of Ischia off the coast of Italy. It is an allusion to the Trojan War on an item nicknamed "Nestor's Cup". This shows that Italians had early exposure to the Greek alphabet, before writing was formalized. Once the Greeks and Italians did standardize writing, they had both settled on the "mirrored" directions.

Edit: You'll notice that the inscription on Nestor's Cup is written from right-to-left, with some letters mirrored to later Greek (nu and epsilon) but others not (sigma).

In those days, there wasn't a set orientation for writing. Some would write left-to-right; others right-to-left. Sometimes the orientation would change direction on each new line (boustrophedon).