In the evolving prayer life of the movement that seeks to renew and transform Judaism and aim ourselves toward healing the world, there have been major experiments with the Alenu. That prayer, close to the end of most services, accepts that it is our double task to celebrate God's creation and to heal its imperfections and injustices.
1. Reb Zalman Schachter-Shalomi, for the Pnai Or/ALEPH siddur-anthology Or Chadash, used "sheh-lo asanu im goyai ha'aratzot, v'lo samanu im mishpachot ha'adamah" [etc etc in the following phrases] -- where the "lo" is lamed-vav, not lamed-aleph, meaning "to Him" or "for Him" -- rather than "not." In order to deal with the false limitation of God to the masculine, two of these four "lo" 's can be said as "lah" -- "for Her" which also signals those who are listening rather than reading that the meaning has shifted. Why do this? To change the basic meaning of para 1 of Alenu, by recognizing that in our era the Jewish people shares a fate with all the peoples of the earth, rather than being divorced from their lot; and to do so in a way that minimally violates the sound and rhythm of Alenu. Some communities use this version on Shabbat and the more traditional one on other days.
2. Judy Chicago wrote a poem which carries the lyrical vision of para 2 of Alenu, which we sometimes chant in English as an analogue. More often we sing [part of] it, to a tune created by Margot Stein-Azen. The song version goes, "And then, and then, both men and women will be gentle; /and then, and then, both women and men will be strong; /and then all will be, /so varied rich & free, that everywhere will be/ Called Eden once again." We sometimes continue with the same tune to move immediately into the Hebrew "Ba'yom ha'hu."
3. We sometimes pause after para 1 of Alenu and ask the kahal to create para 2 by imagining what they see the world "in that day" to be like. People around the circle (we usually davven in a circle, including Alenu & Bar'chu) describe one vivid piece of a world where wickedness has vanished like smoke, etc. For example: once in NYC, someone said, "In that world there would be an Uptown Crosstown subway from Black Harlem to Spanish Harlem to the white East Side -- and no one would be afraid to ride it in either direction." Someone once said, ":In that world when governmental leaders met to make treaties of peace and friendship, it would not be on the front pages -- because it would be so frequent and so obvious." Etc. Sometimes a very heimish image, sometimes very grand -- whatever comes from the circle. We wait for six or seven, sometimes if the flow is strong even more, and then sing "Ba'yom ha'hu."
4. In the second paragraph, the phrase "l'takeyn olam b'malchut shaddai" is usually translated something like "To heal the world under the reign [or kingship] of the Almighty." Taking into account that 'shaddai" is far more likely to stem from "shaddaim," "breasts," I translate it "To heal the world through the majesty oif Nurture.
5. Since some of us use "Yahh" [said to sound like a breath] instead of "Adonai," and "ruach ha'olam" instead of "melech ha'olam," we do this also at the end of Alenu -- "Ba'yom ha'hu yih'yeh Yahh echad," and also (for gender balance) "tih'yeh Yahh achat" and "shma achat." The sense is, "and Yahh will be the breathing/rushing spirit of all the earth."