So, as the title says I'm looking to build a speaker cabinet to use with my bass amp which I'll be using guitar and bass through. Mainly I'm concerned about the need for tweeters and a crossover. Or could I build one with a tweeter that I can possibly turn off? Oh, and I'm not sure about speaker size either. My last bass cabinet I had 12's in, but I didn't use it for guitar much. So, should I go 12's or 10's?
I'm FAT!
If you load it with "bass" speakers it will not get the hi/mids that are pretty essential to guitar, but if you use guitar speakers they may be seriously damaged with bass frequencies.

Why not make a 1x15 for bass and a 1x12/2x12 for use when playing guitar?
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What about a 2x12 cab with one bass and one guitar. trough a head with a bass chanel and a guitar chanel. This, my friend, would kick some serious ass.
there are lots of differences between bass and guitar speakers. Essentially bass speakers have usually pretty flat responses with a 'clean' sound, response goes down to around 40Hz. Guitar speakers have a hiily response with huge mid-range peaks which are part of the sound we associate with electric guitar They don't usually go down below 90Hz and miss at least an octave off . Bass speaker cones are usually heavy and have longer voice coils so bass speakers are much less efficient (quieter ) than guitar speakers.

You are going to struggle to get a speaker to do both jobs well, I suggest you try playing through some borrowed guitar and bass speakers to see if you can live with the compromises before you make an expensive mistake. Try a marshall bass cab if you can get hold of one.

Both Eminence and Celestion do make bass speakers with a prominent midrange 'hump' and you may be able to live with the loss of sensitivity. Only you can decide if you can live with a compromised sound.
electrovoice evm 12's are great for bass and guitar but they sound cleaner than normal guitar speakers in that the speaker itself doesn't distort as much as a clestion or sililar at high volumes. I like it cuz the sound of the amp is more apparent and tighter. mesa boogie used them in the 80's and 90's in a lot of their cabs. i love my 1X12 mesa boogie with the EVM12.
Go a 4 x 12 like you want to do mate. Just have the top half seperate from the bottom so essentially it becomes a 2 x 2 x 12 with seperate input jacks and then you have a guitar on top and bass on bottom or vice versa... That way you get the best of both worlds AND a 4 x 12 cab.