There are two parshios in Titzaveh that seem to be out of place. The bulk of the parsha deals with the garments of the kohanim and their inauguration but the beginning of the parsha discusses the oil for the Menorah and the end discusses the mizbach haketores. The bookends of the parsha seem to fit better in parshas Teruma, what are they doing here?
The possuk in Mishlay (27:9) says שמן וקטרת ישמח לב. The Tanchuma (15) applies this possuk to this parsha that the kohan gadol anointed with oil offered ketores created the atonement for Klal Yisrael on Yom Kippur. The Midrash also says that the Shechina only came to rest in the Mikdash with the offering of the ketores. Says Rav Yitzchak Sorotzkin, that is why the mizbach haketores comes at the end of the description of all the parts of the Mishkan and the kohanim, since it isn't a vessel of the Mkdash but a means to bring about the presence of the Shechina (see Seforno who already alludes to this idea.) What is it about the ketores that brings the presence of the Shechina?
The parsha end off that the Kohan Gadol offers ketores on the mizbaoch on Yom Kippur. Why is this possuk included here when the avodah of Yom Kippur is described in Acharei Mos? The Torah is demonstrating the importance of the avodah of the ketores that it plays a central theme in the avodah of Yom Kippur. The root of קטורת is קטר which is the same word as the Aramic word for tie or connection, קטירא. The ketores represents the deep connection Klal Yisrael has to Hashem. Why is this connection demonstrated through ketores in particular? As discussed in the post 'Smell Of Purity' the sense of smell is the one sense not contaminated by sin and the smell of the ketores represents the internal purity of the person which is not sullied by sin. The Gemarah Berachot (43b) says the neshama benefits from smell. There is no bodily pleasure from smell, it is the neshama, the essence of a person that benefits. This is demonstrated by the fact that a person can be woken up when they are passed out by strong smells for the smell touches the internal life force of the person to awaken the physical body. Rav Moshe Shapira notes that is due to the fact the the ויפח באפיו נשמת חיים comes through the nose of a person, the nose, the sense of smell is the entrance to the attachment between the body and the neshama. He adds that the shape and the word for nose allude to this idea as well. The word for nose is אף, even, something additional and in the physical form the nose protrudes from the face, it something tacked on and elevated from the rest of the face. This is reflective of the nature of the nose to be elevated beyond the normal physicality of the rest of the body. With this idea he explained the Gemarah Berachot (56b) that one who sees an elephant in a dream פיל פילים נעשה לו. The Rema says that we say ומפליא לעשות in asher yatzar because it is a פלא, a wonder that the neshama can be attached to a physical body. We see that פלא refers to the connection of gashmi and ruchni. This is the essence of the פיל and thus it is named, for its nose, its place of ruchni, is its hand, it uses or its physical needs of eating.
אמר רבא חייב איניש לבסומי בפוריא עד דלא ידע בין ארור המו לברוך מרדכי. Rashi explains לבסומי - להשתכר ביין. But why the strange term לבסומי? The word לבסומי is related to the word בסמים, fragrance spices. The Sfas Emes (5640) connects this word with the Gemarah Shabbos (88a) that says when Hashem said the 10 dibros the world became filled with besamim. Why was the world filled with besamim at the giving of the 10 dibros? At the time of Matan Torah the sin of the eitz hadaas was nullified. That is why the dibros came with besamim for they were electric shocks to awaken the neshama. The purity of the untainted part of mankind was awoken.
The Gemarah in Chullin (139b) says that Mordechai is hinted to in the Torah in the words of the ketores, מר דרור ומתרגמינין מרי דכי. Mordechai is connected to the ketores. Ester is also called הדסה, a myrtle known for its nice fragrance. The heroes of the Purim story are connected to smell for they awoke the sleeping souls of Klal Yisrael with the smelling salts to make them realize their predicament. The Sfas Emes concludes that Mordechai is the light of Moshe and hence on Purim also one's drinking is לבסומי. What does that mean? The Gemarah Chullin says Haman is hinted to in the words המן העץ, he is connected to the eitz hadaas. On Purim we connect to the innermost part of our soul untouched by sin. The point of the wine is to unlock the inner depths of a person, to bring out the neshama, the part untouched by the eitz hadaas, by the Haman. That is the antidote to Haman.
Rashi Vayeshev (37:25) says when Yosef was sold to the Arabs as opposed to the normal horrific smelling cargo they normally carried, in this scenario they carried besamim so as Yosef would not have to smell awful smells on his journey with them. It is specifically through the sense of smell that Hashem was indicating to Yosef don't think you are forsaken. I am still connected to your neshama. I am guiding you through this journey. Even though externally it may look bad, internally, from the neshama's perspective, it is for the best.
When one is inline with their nose, with their pure neshama, then there is nothing that stands in the way. There is no Haman, no tumah, that can stand in the way. That is why we find in the ketores, even the חלבנה which smells terrible on its own, is able to be transformed into the pleasant smelling ketores. Rav Yosef Chaim Sonnenfeld (cited by Maharil Diskin) adds that the gematria of חלבנה is the same as המן, they both equal 95. Even the Haman, the foul smell, בני בניו של המן למדו תורה ברבים. The allusion to Mordechai is in the מור דרור which many Rishonim say is a derivative of the blood of a non-kosher animal. Even that which is impure is able to become nullified when its source of kedusha is revealed (see Chasam Sofer and Torah Or Ad D'lo Yadah.) Rav Kook (מדות הראי"ה אות ו) points out it says to wipe out Amalek מתית השמים but in שמים, the root of Amalek can be purified. The Shla (Derush for Zachor) says that is why the holiday of Purim will never be nullified for the entire concept of Purim is about taking away the power of the evil forces and accessing their source of kedusha which is a fulfilment of the mission of the future.
The ketores is burnt between the preparation of the candles of the menorah (Yoma 14b.) Why is it done then? The neros represent the illumination of the souls. The ketores and neros go hand in hand. The neros represent the spread of the light of the neshama outward. The parsha opens with the law of olive oil for the menorah for it is the inner part of the olive, the oil, the inside of a person, the soul is illuminated. The Torah the parsha following the Mishkan highlighting the point of the Mishkan not a mere building but as a point of spreading the light to the world. The parsha ends with the avodah that demonstrates the innermost pure part of a person which is what brings the Shechina to be present in the Mishkan (see Toras Menachem 5752.)