The Gemara in Shabbos 22a says that one should light his Menorah at the entrance of his home on the left side and have his Mezuzah on the right side. The reason given is to be surrounded by mitzvot.
Perhaps another possible explanation as to the significance of having this arrangement to be surrounded by the Menorah and the Mezuzah is as follows.
Regarding the position of the Mezuzah, there is a debate as to whether it should be affixed on the doorpost horizontally or vertically. Rashi holds that the Mezuzah should be placed vertically while Rabbeinu Tam believes that the Mezuzah should be placed horizontally. The Tur concludes that to fulfill both opinions, we compromise and put the Menorah slanted. This idea is supposed to be a message to the household that one should become accustomed to compromise on matters that pertain to the home. If a disagreement erupts between husband and wife, the Mezuzah should remind them of the importance of compromising and coming to a solution that bests suits both parties involved.
The Menorah represents the exact opposite. During the time of Chanukah, the Yivanim tried to get the Jews, to compromise on their Yiddishkeit. They didn’t try to uproot Torah and Mitzvos completely rather they tried to get the Jews to bend and compromise. The victory of Chanukah is not only the victory of the physical battle but also of the spiritual battle that we didn’t compromise no matter what. In essence, the Menorah is a symbol of our unwavering commitment to Torah and Mitzvos.
We surround ourselves with these two mitzvot to guide us. A person must know when he should compromise and give in and when he should remain firm and steadfast in his beliefs!
Have A lichtigeh Chanuka!