NGTS clusters survey $-$ V: Rotation in the Orion Star-forming Complex

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.