The Square Gyrobicupola
The square gyrobicupola is the 29th Johnson solid (J29). It has 16 vertices, 32 edges, and 18 faces (8 equilateral triangles and 10 squares).
The square gyrobicupola can be constructed by joining two square cupolae to each other at their octagonal face, such that the triangular faces of one cupola touch the square faces of the other at the edges. The gyro in the name refers to how the orientation of the two cupolae are gyrated relative to each other. Joining the two cupolae in ortho orientation produces the square orthobicupola (J28) instead.
An octagonal prism can be inserted between the two cupolae, lengthening the square gyrobicupola into an elongated square gyrobicupola (J37).
Projections
Here are some views of the square gyrobicupola from various angles:
Projection  Description 

Top view. 

Front view. 

22.5° side view. 
Coordinates
The Cartesian coordinates of the square gyrobicupola with edge length 2 are:
 (±1, ±1, √2)
 (±1, ±(1+√2), 0)
 (±(1+√2), ±1, 0)
 (0, ±√2, −√2)
 (±√2, 0, −√2)