Charmonium options for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>X</mml:mi><mml:mo>(</mml:mo><mml:mn>3872</mml:mn><mml:mo>)</mml:mo><mml:mn/></mml:math>

In this paper we consider all possible $1D$ and $2P$ $c\overline{c}$ assignments for the recently discovered $X(3872).$ Taking the experimental mass as input, we give numerical results for the $E1$ radiative widths as well as the three principal types of strong decays; open-charm, $c\overline{c}$ annihilation and closed-charm hadronic transitions. We find that many assignments may be immediately eliminated due to the small observed total width. The remaining viable $c\overline{c}$ assignments are $1{}^{3}{D}_{3},$ $1{}^{3}{D}_{2},$ $1{}^{1}{D}_{2},$ $2{}^{3}{P}_{1}$ and $2{}^{1}{P}_{1}.$ A search for the mode $J/\ensuremath{\psi}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ can establish the C parity of the $X(3872),$ which will eliminate many of these possibilities. Radiative transitions can then be used to test the remaining assignments, as they populate characteristic final states. The $1{}^{3}{D}_{2}$ and $1{}^{1}{D}_{2}$ states are predicted to have large (ca. 50%) radiative branching fractions to ${\ensuremath{\chi}}_{c1}\ensuremath{\gamma}$ and ${h}_{c}\ensuremath{\gamma},$ respectively. We predict that the $1{}^{3}{D}_{3}$ will also be relatively narrow and will have a significant (ca. 10%) branching fraction to ${\ensuremath{\chi}}_{c2}\ensuremath{\gamma},$ and should also be observable in B decay. Tests for non-$c\overline{c}X(3872)$ assignments are also discussed.