One place I've seen a similar expression is found in Matthew 6:2, from the Christian Greek Scriptures (New Testament), where Jesus Christ counsels: "...When you go making gifts of mercy, do not blow a Trumpet ahead of you, just as the hypocrites do in the synagogues and in the streets, that they may be glorified by men...". While the wording we use daily has changed a bit since this text was written near 41 CE, it implies the same thing. If we are "tooting our own horn" or "blowing a trumpet ahead of us" we would be boasting or bragging about our achievements or some aspect of our life, which could have a negative affect on others (ie. jealousy, envy). Today, though, the phrase is definitely used a bit more lightly, and with laughs.
A palindrome for blowing a horn could be: 'toot'.
The palindrome word for the sound of a horn is "toot."
A palindrome for the sound of a horn is "toot."
toot
toot
Toot
toot
toot
toot
toot
"Vroom" is a palindrome for the sound of a horn.
toot