How do I reorder tmux windows?

In screen, I can just type C-a :number 0 to move a window to the top of the window list and push all the other windows down one. What's the equivalent command sequence for tmux? I looked at the man page, but I'm finding it confusing on this point.

The swap-window command is closest to what you want.

"Prefix :" (that is Ctrl+B, : by default) brings you to the tmux-command prompt. There you enter:

swap-window -s 3 -t 1

to let window number 3 and window number 1 swap their positions.

To swap the current window with the top window, do:

swap-window -t 0

In the unlikely case of having no window at index 0, do:

move-window -t 0

(if base-index is 0, as it is by default). Command move-window -t <NUMBER> is by default bound to Ctrl+B, ..

You can bind that command to a key (T for "top" for example) by adding the following to your ~/.tmux.conf:

bind-key T swap-window -t 0

Adding to Gareth's answer, you can use the following key bindings

bind-key -n C-S-Left swap-window -t -1
bind-key -n C-S-Right swap-window -t +1

Pressing Ctrl+Shift+Left (will move the current window to the left. Similarly right. No need to use the modifier (C-b).

For tmux 3.0 version, you should use following key bindings

bind-key -n C-S-Left swap-window -t -1\; select-window -t -1
bind-key -n C-S-Right swap-window -t +1\; select-window -t +1