SharePoint 2007 - Cannot access "Page Settings" area in Publishing site

Here's an odd thing. I wanted to change the Page template of a SharePoint page that was displaying in an old, outdated template. But when I went Edit Page, then Page > Page Settings ... 

... I wasn't seeing the familiar Page Settings page. I was seeing this:

No sign of my picklist of Page Templates at all. I knew I had encountered this problem before, but couldn't remember what the solution was, so I got on to trusty Google, figuring someone is bound to have the answer to this vexing issue. But no ... nothing doing.

I tried going to the Pages Directory and seeing if Editing Properties of the page would help, but that got me nowhere.

Then, running out of options, I fired up SharePoint Designer and navigated to the sub-site. And there it was! 

The dad-blasted Page Layout was detached, obviously from some earlier Designer fiddling and I'd forgotten to reattach. Simply reattaching the Page Layout fixed the problem and I was able to access the Page Layout picklist in the Page Settings area.

It might seem screamingly obvious but then, you can't always remember everything, and incredibly, no one has posted an explanation for this problem that crops up in the first ten screens of a Google search. 

So hopefully, this will save someone thrashing around, trying to figure out why they can't change the Template for a SharePoint Publishing page.