What is the most precise classification of the quadrilateral formed by connecting the midpoints of the sides of a square?

We have to form a quadrilateral by connecting the midpoints of the square.
The side of the square: a.
The side of the new quadrilateral: x² = (a/2)² + (a/2)²
x² = a²/4 + a²/4 = a²/2
x = a√2 / 2
All sides of the quadrilateral are equal: a√2 / 2.
So it could be another square or a rhombus. But all interior angles of the new quadrilateral are the same: 90°.
This is the square.