During polarity reversals of the Earth's internal geomagnetic field, the paleomagnetosphere must have undergone dramatic changes because the core field was essentially different from the present-day case, where the dipole moment is approximately aligned with the Earth's rotation axis (axial dipole) and perpendicular to the solar wind flow. According to one of the possible transition scenarios, the internal dipole moment, having much reduced in magnitude, could slowly turn around, staying close to the equatorial plane for a significant time. In this paper we present magnetohydrodynamic (MHD) simulations of a so-called equatorial dipolar paleomagnetosphere, where the internal dipole moment is perpendicular to the rotation axis, i.e., the dipole axis is in the equatorial plane. The magnitude of the dipole moment was chosen as small as one-tenth of the present value. With such a dipole field orientation, the dipole tilt in GSM coordinates changes, due to the Earth's rotation, between zero and 360 degrees in the course of the day, resulting in an extremely dynamic magnetosphere on the diurnal timescale. As a first approximation, we calculated steady-state solutions with different dipole tilts, or in other words with different angles between the dipole axis and the Sun-Earth line, to represent the paleomagnetosphere at different times of the day. We describe the regular diurnal variation of the geomagnetic field-line configuration and the topology of large-scale current systems, like the magnetopause currents and the tail current sheet. We investigate the so-called pole-on magnetosphere, where the dipole axis is aligned with the Sun-Earth line, in more details using different solar wind input parameters.