Curious congruences for cyclotomic polynomials

Let ${\mathrm{\Phi }}_n^{(k)}(x)$ be the $k$-th derivative of $n$-th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: $2{\mathrm{\Phi }}_n^{(3)}(1)$ is divisible by $\varphi (n)-2$ and ${\mathrm{\Phi }}_n^{(2k+1)}(1)$ is divisible by $\varphi (n)-2k$ for $k\ge 2$. The congruence stems from a general property of self-reciprocal polynomials.