Hmmm. The plot thickens. I booted up another computer, HOMEOFFICE2, and tried to ping BIKEDESK. HOMEOFFCE2 couldn't get a response from BIKEDESK either.
But, again, BIKEDESK could ping HOMEOFFICE2. So, in both cases BIKEDESK can ping out, but other machines can't ping in. Obviously RDP was a...