How does the PNP transistor in this IR receiver circuit work?
So you're thinking that the PNP is "upside down"?



I agree with you, note how the arrow in the emitter points in the "wrong" way.

However A PNP still works as a PNP like that!

Only it will have a much lower current gain. Here that's not such an issue as the current through R1 will be larger than the current through R2.

If normally \$\beta\$ (the current gain) is around 200 then in this case (emitter and collector swapped) it will be a lot less like 5 or worse. Then there's basically very little current gain.

In this circuit that is not such an issue as the output of the IR sensor is (probably) an open collector (NPN) output meaning that it pulls to ground. That would discharge C1. C1 is charged (when the IR sensor's output is not pulled low) via R1 and the PNP. That current will also result in a current flowing out of the emitter of the PNP (this emitter now behaves as a collector!) into R2 and that will provide the clock to the 4017's input.

So I agree, looks weird. But does (appear to) work according to that website and comments. I have not tried it myself though.