#Husk , 6 bytes

mΣ∂m↔T

Try it online.

Explanation:

 T # Transpose the (implicit) argument; swapping rows/columns
 m↔ # Reverse each row
 ∂ # Take the anti-diagonals of this matrix
mΣ # And sum each inner anti-diagonal list
 # (after which the result is output implicitly)

Husk , 6 bytes

mΣ∂m↔T

Try it online.

Explanation:

 T # Transpose the (implicit) argument; swapping rows/columns
 m↔ # Reverse each row
 ∂ # Take the anti-diagonals of this matrix
mΣ # And sum each inner anti-diagonal list
 # (after which the result is output implicitly)