I have two adjacent ceiling fixtures holding two 40W 48" fluorescent
tubes, both controlled by a wall switch. Sometimes, they go on
properly; other times they'll go on if their glass tubes are touched;
still other times they need to be rotated slightly in their sockets.
And sometimes they won't go on at all.

What should I check?

The fixtures are at least 18 years old, but do not have the little
starter cans that older fixtures used to.