We present a study of rotation across 30 square degrees of the Orion Star-forming Complex, following a $\sim$200 d photometric monitoring campaign by the Next Generation Transit Survey (NGTS). From 5749 light curves of Orion members, we report periodic signatures for 2268 objects and analyse rotation period distributions as a function of colour for 1789 stars with spectral types F0$-$M5. We select candidate members of Orion using $\textit{Gaia}$ data and assign our targets to kinematic sub-groups. We correct for interstellar extinction on a star-by-star basis and determine stellar and cluster ages using magnetic and non-magnetic stellar evolutionary models. Rotation periods generally lie in the range 1$-$10 d, with only 1.5 per cent of classical T Tauri stars or Class I/II young stellar objects rotating with periods shorter than 1.8 d, compared with 14 per cent of weak-line T Tauri stars or Class III objects. In period$-$colour space, the rotation period distribution moves towards shorter periods among low-mass (>M2) stars of age 3$-$6 Myr, compared with those at 1$-$3 Myr, with no periods longer than 10 d for stars later than M3.5. This could reflect a mass-dependence for the dispersal of circumstellar discs. Finally, we suggest that the turnover (from increasing to decreasing periods) in the period$-$colour distributions may occur at lower mass for the older-aged population: $\sim$K5 spectral type at 1$-$3 Myr shifting to $\sim$M1 at 3$-$6 Myr.
Comment: This article has been accepted for publication in Monthly Notices of the Royal Astronomical Society. Published by Oxford University Press on behalf of the Royal Astronomical Society. 20 pages. 21 figures