Now, this is probably just my naive view on the matter, but wouldn't it be relatively easy to lift all the pikes of the formation, turnabout 90 degrees and lower them again to face the enemy?

You might think that, but formation evolutions are quite a bit more complicated than you'd expect. More can always be said on the matter if anyone would like to address themselves to the question; for the meantime, as this is one of the more common questions we get, I commend to your attention this previous thread on the matter, where u/Iphikrates comments on the Greeks, while u/hborrgg addresses Early Modern pike formations, with much similar considerations.