The Sun-Dial Theory
Were they for progressive meditation and praise as worshippers approached the temple on pilgrimmage? Were they specifically for the priests and levites in approaching holy orders? Were they a celebration of return of the exiles under Zerubbabel or Ezra or Nehemiah? Did they correspond to a ritual of ascending fifteen literal steps constructed in the temple (but not as of the days of King David)?
One can see from the text that 5 are attributed to David. Ten remain anonymous.
Another theory attributes the 10 to King Hezekiah of later date. Certainly he was in distress as the Assyrians swarmed his kingdom and as he suffered from a supposedly incurable disease. He turned his face to the wall, laid out the threatening letter of his enemies and wept. His fervent prayer and the intercession of the prophet Isaiah reversed all of this. The King was given a sign that rescue was forthcoming; the sun-dial would be turned back 10 degrees. The victory did come, and the healing, and shortly thereafter an end to the King's childlessness. All of these facts are suggestive of portions of the Psalms here considered.
See Isaiah 38:
7And this shall be a sign unto thee from the LORD, that the LORD will do this thing that he hath spoken;
8Behold, I will bring again the shadow of the degrees, which is gone down in the sun dial of Ahaz, ten degrees backward. So the sun returned ten degrees, by which degrees it was gone down.
19The living, the living, he shall praise thee, as I do this day: the father to the children shall make known thy truth.
20The LORD was ready to save me: therefore we will sing my songs to the stringed instruments all the days of our life in the house of the LORD.
I do not bring a verdict on this sun-dial theory, but I do invite a look again at the ten psalms in light of the Hezekiah scenario (120, 121, 123, 125, 126, 127, 128, 129, 130, 133). Interesting perspective. An impossible situation turned around by the mercy of God. (Perhaps to be compared to David at Ziklag or Peter in prison awaiting execution.)