HQs can fight, and do damage, if you attach a front line brigade to them. The main advantage to attaching a line brigade is that the commanding officer will gain experience not only through the combats of his subordinate divisions, but through the direct actions of his own HQ division as well. A secondary consideration is that it provides one more division to the Corps, giving you more divisions on the map without requiring more commanding officers and HQs. The tertiary effect is that it slows the HQ down to match the rest of the Corps.
I consider the attached units as "Corps Assets" (recon, snipers, additional AT and ART, communications, and other specialists of various sorts), not that the HQ staff personnel are taking turns on the front line. The HQ has actual attack values (both Hard and Soft attack), which can be further boosted through doctrines in the mobile group, so by keeping it behind the lines, you're paying the manpower, officer ratio, and supply costs for a unit that never uses its firepower.
The "reorganize" command doesn't work directly on lone HQs, but you can select the HQ and a brigade you wish to attach, and press the "r" key to bring up the reorganization screen, or else select it and a division with more than one brigade. Freshly built brigades can be assigned to the HQ from the deployment screen as well. Once you have another brigade attached to the HQ, the "Reorganize" command functions normally.