Using spectral-line observations of HNCO, N2H+, and HNC, we investigate the kinematics of dense gas in the central ~250 pc of the Galaxy. We present SCOUSE (Semi-automated multi-COmponent Universal Spectral-line fitting Engine), a line fitting algorithm designed to analyse large volumes of spectral-line data efficiently and systematically. Unlike techniques which do not account for complex line profiles, SCOUSE accurately describes the {l, b, v_LSR} distribution of CMZ gas, which is asymmetric about Sgr A* in both position and velocity. Velocity dispersions range from 2.6 km/s28. The gas is distributed throughout several "streams", with projected lengths ~100-250 pc. We link the streams to individual clouds and sub-regions, including Sgr C, the 20 and 50 km/s clouds, the dust ridge, and Sgr B2. Shell-like emission features can be explained by the projection of independent molecular clouds in Sgr C and the newly identified conical profile of Sgr B2 in {l ,b, v_LSR} space. These features have previously invoked supernova-driven shells and cloud-cloud collisions as explanations. We instead caution against structure identification in velocity-integrated emission maps. Three geometries describing the 3-D structure of the CMZ are investigated: i) two spiral arms; ii) a closed elliptical orbit; iii) an open stream. While two spiral arms and an open stream qualitatively reproduce the gas distribution, the most recent parameterisation of the closed elliptical orbit does not. Finally, we discuss how proper motion measurements of masers can distinguish between these geometries, and suggest that this effort should be focused on the 20 km/s and 50 km/s clouds and Sgr C.

Accepted for publication in MNRAS. 30 pages (including appendices), 22 figures, 3 tables. This updated version includes minor corrections (including typos and updated references). SCOUSE is available for download at https://github.com/jdhenshaw/SCOUSE. A selection of PPV movies can be downloaded at https://github.com/jdhenshaw/PPV_movies