Now I don't know what to think :(
Try "I should trust Ran's interpretation, because his adapter works, but mine doesn't"
It's easy to see why you might've gotten the wrong impression, since most of the connectors you've seen number sequentially down the rows. Header connectors are different because they're usually used with ribbon cables, so someone decided to use the "even-odd" scheme to make the pin numbers correspond to the conductors in the ribbon. I dunno whether there's an official industry standard for it, but I've seen dozens of designs numbered that way over the years, so my automatic assumption was that this one was done the same way.
I'm using an adapter that I bought as a kit some years back, and plug into the projet-du-jour with a cable to match the project's pinouts. It has a DB-25 soldered to the board, so I just made up a 9-to-25 pin cable with what I thought were the right crossovers and loopbacks.
Specifically, I wired up the Webpal side so it looks modem-ish, because my adapter has a female connector, and I only plan to connect to the PC (and, btw, this resulted in one of my earlier comments being misleading: when I said the webpla "dropped CTS", it was actually the signal bigbrd labels "RTS" on his schematic. It just looks like CTS on the adapter's DB-25).
I looped DTR back to DSR and DCD on the PC end, and left the handshake inputs to the webpal open, just like bigbrd did.
I crossed CTS and RTS (as labelled by bigbrd), null-modem style. This seems to have caused the problem mentioned in my last post, even though it's the right way to do it for proper hardware handshaking (shrug). If it turns out that the protocol really needs hardware handshaking, I guess I'll have to add a switch to get it past the initialization problem. But, since no one else has mentioned it, I'm guessing I can just leave it wired as in the "fake-out box". At 9600 baud, it's unlikely that hardware handshaking is really needed, anyway.
I also made sure that TD and RD wre going in the right directions: bigbrd's pinout is kind of a mix of DTE and DCE conventions, so I used the "in vs out" of the buffer chip as the definitive source.