Does a comet’s tail (or lack there-of) tell us what direction it’s heading?

When Comet Holmes was heading away from Earth, we on Earth didn't see an angled view of the comet's tail. Was it tailless because of its motion? That is, would a comet also appear tailless if it were approaching, with Earth right in its path?

Comet Holmes

Comet Holmes on the night of October 25th as captured by Gil Esquerdo using the Smithsonian Astrophysical Observatory's 48-inch reflector at Whipple Observatory in Arizona.

It's impossible to tell where a comet is heading based on the way its tail points. A dust tail is produced by microscopic dust particles released from the comet by the Sun's heat and driven away by sunlight (solar radiation pressure). So in 3-D space a comet's tail usually points away from the Sun, whether the comet is inbound or heading out.

This is why a comet low in the western evening sky at dusk (or low in the east before dawn) shows a tail extending generally upward, away from the Sun's location below your horizon.

— Roger W. Sinnott