Lord Ganesha is considered a god of wisdom and prosperity. He is worshipped first before the inception of any auspicious work in Hinduism. Everyone like to have wisdom and prosperity, hence everyone love to have the blessings of elephant headed lord Ganesha. But all the images of Lord Ganesha are not considered auspicious to keep at home.
The image or idol in which his trunk is inclined his right is not considered auspicious to keep at home. It is advised to keep outside or keep in certain temples. The image or idol in which his trunk is inclined left is considered auspicious and advised to keep at home and worship.
Why is the same god treated differently due to his different posture? What is the logic behind it?
As per the legends, Ganesha was created by his mother Parvati without the involvement of lord Shiva. She is known as Anapurna as well. His head was chopped off by Lord Shiva. Later, he was resurrected by lord Shiva by replacing his head with an elephant one.
Now the existence of Ganesha is due to both Shiva and Parvati. Parvati contributed his body(matter) and Shiva contributed his head(wisdom). Shiva is an ascetic who lives in a secluded place far from the materialistic world. Parvati is Annapurna, without her the existence of the world is not possible. Parvati is suited for the householders and Shiva is for hermits. Parvati forces to participate in the world and Shiva withdraws from the world. Parvati is for entanglement in "Samsara", the cycle of birth and death, and pain and pleasure. Shiva`s path is for getting rid of the "Samsara" and move to salvation.
A strict penance is required for getting the permanent rid of the duality (birth and death, pain and pleasure) of the "Samsara". Even a small mistake in the path of salvation creates a big obstruction. This is the path of the hermit, who seeks the meaning of life and death rather than perplexed by it, and gets rid of all kinds of suffering occurs due to the illusion of the world. Hermits are not controlled by the matter, rather matters are controlled by them. They are not influenced by the alluded attraction of the materialistic world.
For the householder, following strict regulations of the worship and rituals are not possible, there may be some mistakes somewhere. Small mistakes are acceptable for the householders. Householders make this world sustainable by producing the food and procreation. And they are supported by Anapurna/Parvati, the goddess of the materialistic world.
Scientifically, the right side of the body is controlled by the left side of the brain and vice-versa. The left brain is the logical brain and the right brain is the emotional brain. And also the side of the heart is left. Hermit acquires Yogic power, logic, and intellect by their unerring penance to understand the whole existence. Without understanding the existence as it is, the salvation of the continuous birth and death is not possible. A hermit is not allowed to do any mistake knowingly or unknowingly or driven by their emotions.
Parvati is "Vamini" of Shiva. Vamini means who sits left side - wife. Lord Ganesha sits in the middle of his parents. If his trunk is inclining right, means he is supporting the quality of the hermit. He is on his father side blesses his devotee of the salvation. He is Siddhivinayak. If his trunk is inclining left, he is supporting the quality of the householder. Being at his mother`s side, he blesses his devotees of the prosperity to the run the wheel of this world hurdle free. He is Ikshapurti.
That is the reason - left curled trunk idol or image of Ganesha is advised for the householder, not the right curled trunk.