Yeah, you can. Just been round this buoy, as my son's learning the clarinet en route (he currently thinks) to the sax, when his hands are big enough. His tutor says not before Grade III, so we'll see. Have an Alto hiding in the loft for him anyway.
I started with playing the Sop to help him along with lessons and practice, but it's not an exact match really, save for being in the same key and overlapping in range to some extent, and in particular the fingering's so dissimilar as to be confusing for him to follow. So, picked up a perfectly decent used clarinet for well under a hundred quid, and off we went.
The fingering is a little more complicated/different to the sax, and the Bb clari has the largest range in the woodwind family (I'm pretty sure I'm right in saying) so there's more of it, and more alternative fingerings, but it's not chalk and cheese by any means. Playing over the break is a bigger deal than with a sax. Embouchure, or at least correct embouchure, is quite different, but that won't stop you getting a half decent sound out of it pretty much off the bat.
You can do some interesting stuff you can't do on the sax - huge glissando (think opening of Rhapsody in Blue) coz you can slide your fingers off the holes as well as bending the pitch with the reed. I have to say I'm not the greatest fan or the sound of the clarinet, but somewhat to my surprise, I'm finding it fun to fool around on.
Just be careful you don't start wearing stripey waistcoats and bowler hats...