Lord Ganesha has one Tusk.
One Tusk remains broken.
The common legend I have been aware that Lord Ganesha broke hos tusk to write the Mahabharata at the bidding of Veda Vyasa.
There is another legend in Mudugala Purana.
Parashuarama is a Chiranjeevi, who lives forever, went to meet Lord Shiva , His Teacher in the Himalayas and was blocked by Ganesha who was guarding Kailasa.
Angry Parsashurama threw his axe at Ganapati and it broke His left Tusk.
Ganesha accepted this without making effort to stop it as the Axe was given to Parashurama by His father Shiva .
Furious Parvati, hurt at Ganesha being hurt,declared she would cut off the arms of Parashurama.
She took the form of Durga, becoming omnipotent, but at the last moment, Shiva was able to pacify her by making her see the avatar as her own son.
Ganesha without one tusk is considered as His second avatar by some.
Ganesha Gayatri gives importance to this aspect of single Tusk
‘Ekadanthaaya Vidhmahe Vakra Dhundaaya Dheemahi,
Thanno Danthi Prachothayaath’
Source.Mudugala Purana,which is classified as a Upapurana.