Shaivam - A way of life.
Ganapathy - Vision and Initiative
Vision is something that is not present today but picturize it so that you will achieve that picture one day. For example, Right now there is a lot of poverty in the country, you will have a vision that one day there is no poverty that everyone get sufficient food, water and shelter.
Vision is a picture that is not real during the present time but will be in the mind for the future and you take initiative towards that.
Ganapathy is a representation of Vision and Initiative, therefore we have a human body with elephant head to symbolize something that is not real for now but you still get picture of it. Ganapathy is also the symbol of initiative and as you take initiative you will be removed from all obstacles automatically.
Lord Subramaniam - Local importance and destroying internal evil.
Local importance is anything that gives importance to self and any associations with self. It could your locality, your family, or any associations.
Lord Subramanian (or Lord Murugan) is overwhelmingly popular among people of Tamil Nadu for this reason, the urge for their locality to be best be it arts, literature, science, technology and other fields.
Lord Subramanian is also a symbol of destroying internal evil (internal to one's own mind) that arises by compromising self and its associations or simply compromising local importance.
Parvathy - Destruction
It is a common misconception that Shiva is the destroyer of evil, actually destroying evil is the responsibility of Parvathy. If there is anything that disturbs the peace of a society then Parvathy takes charge. If Parvathy start the thaandavam even Shiva cannot stop her.
Shiva - Creator and sustainer of the nature.
"Naan asainthaal asaiyum akilamellame" - The world is according to his movements, the nature progresses according to his plan and the nature sustains according to his design.