We present a global-in-radius linear analysis of the axisymmetric magnetorotational instability (MRI) in a collisional magnetized plasma with Braginskii viscosity. For a galactic angular velocity profile Ω we obtain analytic solutions for three magnetic field orientations: purely azimuthal, purely vertical and slightly pitched (almost azimuthal). In the first two cases the Braginskii viscosity damps otherwise neutrally stable modes, and reduces the growth rate of the MRI, respectively. In the final case the Braginskii viscosity makes the MRI up to times faster than its inviscid counterpart, even for asymptotically small pitch angles. We investigate the transition between the Lorentz-force-dominated and the Braginskii viscosity-dominated regimes in terms of a parameter where νB is the viscous coefficient and B the Alfvén speed. In the limits where the parameter is respectively small and large, we recover the inviscid MRI and the magnetoviscous instability. We obtain asymptotic expressions for the approach to these limits, and find that the Braginskii viscosity can magnify the effects of azimuthal hoop tension (the growth rate becomes complex) by over an order of magnitude. We discuss the relevance of our results to the local approximation, galaxies and other magnetized astrophysical plasmas. Our results should prove useful for benchmarking codes in global geometries.