Michael's number

Algebra Level 3

For a positive four-digit integer $n,$ let $T(n)$ be the number created by swapping the hundreds and thousands digits of $n$ and swapping the tens and units digits of $n.$ There is a unique integer $M$ such that $T(M)= 4M.$ What are the last three digits of $M$ ?

This problem is posed by Michael T.

Details and assumptions

For example, $T(1234)=2143$ and $T(1508) = 5180.$