Thanks a lot for your input, guys!
To clear some points up:
+ I will be using an 16F877A PIC, since it is either that or a 18F (I have nothing else lying around here). So PWM could be an option (I must confess that it looks "elegant" - jsut a matter of taste, of course).
+ The first signal was measured using a 36 kHz decoder (which was sold to me as a 38 kHz one!). I'm now waiting for the correct one to arrive at my mailbox...
+ The signal I've posted was sampled using a photodiode. So we have carrier plus data.
+ The PIC will be doing other things but, even though I, as a matter of principle, do not like blocking code on a non multitasking system, it is OK for it to stay "blocked" for the 200 ms or so that the transmission seems to last.
+ Sceadwian is right, I don't need to decode the bitstream, just put the same signal out to control this damn A/Cs (my wife just "likes" to leave them on when she leaves for work - actually, even I forget to turn the three things, in different rooms, off). Understanding the coding method would be a nice bonus for me, but in no way essential.
Thanks again, I now think that with your help I'm on the right track!