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We propose a classification of the reflection $K$-matrices (solutions of theboundary Yang-Baxter equation) for the$U_{q}[\mathrm{osp}^{\left(2\right)}\left(2|2m\right)]=U_{q}[C^{\left(2\right)}\left(m+1\right)]$vertex-model. We have found four families of solutions, namely, the completesolutions, in which no elements of the reflection $K$-matrix is null, theblock-diagonal solutions, the $X$-shape solutions and the diagonal solutions.We highlight that these diagonal $K$-matrices also hold for the$U_{q}[\mathrm{osp}^{\left(2\right)}\left(2n+2|2m\right)]=U_{q}[D^{\left(2\right)}\left(n+1,m\right)]$vertex-model.

Reflection matrices with $U_q[osp^{(2)}(2|2m)]$ symmetry