Based on a hat I saw in Gamla Stan, the old part of Stockholm, with a tvåändsstickning (Swedish twined knitting) brim and stranded cap.
ETA 12/9: the links didn’t work in the original PDF, so I’ve now updated it to fix them. Sorry!
Note: Because someone asked, the gauge given is for the box-patterned, upper part of the hat. The gauge at the brim is more like 26 sts / 4”; the hat edge measures about 20” (It’s possible the lining I sewed in is stretching it a bit.)
12/11: Updated pattern to add some sizing information and correct one mistake: you need an odd number of stitches for the tvåändsstickning section, but an even number for the box pattern, so you need to M1 at the end of the brim.
12/15: Found and corrected error in the tvåändsstickning section. CO should be 136 sts after all, and colors in twined knit rows should be lined up with those in crook rows.
12/18: Added full details in metric as well as English, clarified box pattern chart.