How to get this list with a terse method

The following approach will be very fast for large lists since it utilizes vectorization:

Table[{a, b}, 4] + Join[#, -#] &[IdentityMatrix[2]]

Just for fun:

Join @@ (# + {a, b} & /@ {#, -#} & /@ {{1, 0}, {0, 1}})

A double Transpose might be considered "beautiful", and it certainly can be very fast too.

Transpose[Transpose[{{-1, 0}, {1, 0}, {0, -1}, {0, 1}}] + {a, b}]