It's possible to watch past games on Tenhou. Games might never actually be deleted (needs verification). There's various ways to watch past games (and not only your own).
On the lobby menu there's a replay option (top one). This gives you an intricate list of past games, dates, scores players and other wonderful features. In particular, it gives you WWW URLs to give to other people to watch (or watch for yourself).
Note! The list is stored on the computer. While the actual game records are server-stored, if you clear private settings and records these lists will also be lost.
The first tab is Inspect. It has a brief list of all the games with links (the blue text) to watch in the currently loaded client or in other ways. The numbers in the square braces  go in the format mm/dd, followed by a link to the game and a list of the players in each particular game with finishing points in braces (). Quit or disconnected games have blank scores because the computer wasn't there to see the outcome, although you'll see it if you go through the game.
This droplist picks the way in which the blue links in the Inspect tab act. Top to bottom, the modes are:
The next button clears all computer stored record lists. Pressing this and confirming it will delete all locally stored records. Though they are still accessible you won't have a list of them (and thus is is pretty hard to find them, stored or not).
The last far right button closes the window.
This is the Diary tab. It has a chronologically ordered list of games played (numbered in square braces 【】). The position in which you came followed by your score and the game type is next. Below each entry is a WWW URL which you or anybody else can use to watch the game. The figures at the top are: games played; sum of points; average points; (unknown); respectively.
The last tab, Backup, is a plain text list of all games played. The format of each entry is:
[yyyy/mm/dd/game type] A:You(points) B,C(,D):other players
URL to game replay
This list is ideal for parsing into some kind of computer program if need be - a real backup.
This tab is similar to all the others in the log window. Links default to opening in the same window, though.
Whatever way you pick to get replays, the replay controls are all in English, so it's all self-explanatory. To advance manually, click anywhere on the screen.