Ganesha Idols exhibits both, right handed and left handed trunk.
Right handed trunk
These idols are called Dakshinamurthy or Dakshinabhimukhi, which means they are south facing. South direction leads to Yamalok (the abode of Yam, Lord of Death) while right direction symbolizes Surya Nadi (The Solar Channel), which gives radiant glow to the person who has activated it.
It's for this reason, such Ganesh Idols are caled Jagrut (active).
Hindujagruti says about right handed Ganesha Idols not worshipped,
The dakshinabhimukhi idol is not worshipped ritualistically in the usual manner because tiryak (raja) frequencies are emitted from the south. The ritualistic worship of this idol is performed by observing all the norms of ritualistic worship meticulously. Consequently the sattva component is augmented and one is not distressed by the raja frequencies coming from the south.
Left handed trunk
These murties are called Vamamukhi, which means North facing. Left signifies Chandra Nadi (The Moon Channel) which signifies bliss and Ananda (happiness). This idol can be worshipped ritualistically.