Phys.Rev.D65:052008,2002 The D0 Collaboration has studied for the first time the properties of hadron-collider jets reconstructed with a successive-combination algorithm based on relative transverse momenta ($k_{\perp}$) of energy clusters. Using the standard value D = 1.0 of the jet-separation parameter in the $k_{\perp}$ algorithm, we find that the $p_T$ of such jets is higher than the $E_T$ of matched jets reconstructed with cones of radius R = 0.7, by about 5 (8) GeV at $p_T \approx 90$ (240) GeV. To examine internal jet structure, the $k_{\perp}$ algorithm is applied within D = 0.5 jets to resolve any subjets. The multiplicity of subjets in jet samples at $\sqrt{s} = 1800$ GeV and 630 GeV is extracted separately for gluons ($M_g$) and quarks ($M_q$), and the ratio of average subjet multiplicities in gluon and quark jets is measured as $(M_{g} - 1) / (M_{q} - 1) = 1.84 \pm 0.15 (stat.) \pm ^{0.22}_{0.18} (sys.)$. This ratio is in agreement with the expectations from the HERWIG Monte Carlo event generator and a resummation calculation, and with observations in $e^+e^-$ annihilations, and is close to the naive prediction for the ratio of color charges of $C_A/C_F = 9/4 = 2.25$.