  We present two novel results about Hilbert space operators which are
nilpotent of order two. First, we prove that such operators are indestructible
complex symmetric operators, in the sense that tensoring them with any operator
yields a complex symmetric operator. In fact, we prove that this property
characterizes nilpotents of order two among all nonzero bounded operators.
Second, we establish that every nilpotent of order two is unitarily equivalent
to a truncated Toeplitz operator.
