He shall take a shovelful of fiery coals from atop the Mizbeyach that is before Hashem, and two handfuls of finely ground incense, and bring it within the Curtain. (16:12)
The Mishnah in Maseches Yoma relates that the Kohen Gadol would enter the Kodesh Hakodashim carrying the shovel with the coals in his right hand, and the ladle with the two handfuls of ketores in his right hand. The Gemara elsewhere notes that this represents a unique situation in the Beis Hamikdash. As a rule, avodah is only valid when performed with the right hand. The one exception is our case of Yom Kippur, when the ketores is brought in with the left hand.
What is the significance behind the left hand being included in the avodah of Yom Kippur specifically?
The Meshech Chochmah explains that the right side represents the person’s intellectual faculties, while the left side represents the more natural part of his makeup. This is the meaning of the statement in the Gemara, “Regarding one’s desire, his child and his wife, the left hand should push them away and the right hand should bring them close.” This means that when it comes to one’s basic and natural needs, he should push away the left side, not allowing his natural faculties to engage in these matters entirely unchecked; rather, it is the right side that should bring them close, his involvement with them should be guided through wisdom and higher perspective.
For this reason, the avodah in the Beis Hamikdash is only ever performed with the right side, for it is specifically one’s higher faculties of wisdom and reason that are appropriate to involve in the Divine service that takes place there. The exception to this rule is Yom Kippur. On that holy and sublime day, through the fasting and other abstinences from physical pleasure, the person becomes detached from and elevated above his base physical existence, becoming likened to an angel. In this lofty state, even the “left side,” i.e. his natural physical existence, becomes elevated to become a worthy participant in the avodah of the day.
