Why does my stockbroker allow trading at the close but not at the open?

If I understand everything correctly, TD Ameritrade does support market-on-open orders. Simply submit a market order while the market is closed, and it will be executed as a market-on-open order when the market opens.

In order to try to verify this, I looked at the fill prices of a couple of market orders I placed while the market was closed. I looked at two buy orders on different days, and both of them executed at that day's opening price listed on Yahoo Finance. I also looked at one sell order, and it executed at a price one cent better (higher) than that day's opening price.

It also seems to be possible to do a limit-on-open order by submitting a limit order with a fill-or-kill instruction while the market is closed. Presumably, when the market opens, if the opening price satisfies the order, then the order will be filled at that price, and if the opening price does not satisfy the order, then the order will be canceled.

Just now, I verified that it is possible to place such an order, but I have not confirmed that such an order will actually behave as expected.