Yes but are the avast processes in my image (click to enlarge it) running.
If the icon is present fine, when it is rotating it basically shows that something is being scanned so is a reasonably quick indication that avast is working.
In XP some icons could be hidden, by clicking the < arrow to the left of the clock it opens up the tray area.
Some other security software can kill the avast icon (though the avast self-defence module should try to prevent that.
What other security software do you have AdAware, Spybot Search & Destroy, Windows Defender, etc. ?