To add a datapoint: There's a lot of really fun "office appropriate" cycling wear for commuters these day. You don't need to wear lycra. I know many triathletes who commute to work by bike and don't wear their race kits to work, they wear commuter gear. My DP has cycling pants that look like they came from a suit. With a suit jacket and tie, you'd never know they had flat seams inside and an elastic gusset in the crotch. DP can wear those pants to the courthouse and client meetings without anyone realizing they are cycle wear.
The clothes might look cute, but won't you be all sweaty from biking? I can't imagine sitting a cubicle over from someone that biked in and didn't change or anything and having to smell stale sweat wafting over the cube wall.
Of course I change my clothes when I get to work! We have horrendous humidity up here too and it would be utterly uncomfortable not to, and I'd end up smelling like a bear by lunchtime. The difference is that I don't treat the ladies restroom as a locker room. As I mentioned upthread, I can change quickly in a bathroom stall or lock my office door (*sigh* I have no window). Bathing in the bathroom sink or walking around the restoom in the nude is completely inappropriate in an office setting -- it's a corporate workplace, not a gym.
The point of the appropriate cycle wear is that here are many options that can blend in to a general office setting, so you can arrive and leave without being disruptive and making a spectacle of yourself.
People have also figured out all kinds of different solutions when showers aren't available too. I have a mist bottle with a dilute alcohol/water mix. A quick mist and wipe down with a towel before I re-apply deoderant and put my fresh clothes on and I'm dry as a bone and odor free. My colleagues who walk from the subway station to our building, however, remain sweat covered. (Except for the one guy who also brings in a fresh shirt because he has a 20-minute walk to work from his apartment.)
Edit: However, as someone with long hair, "helmet hair" is something that continues to vex me.