Relativistic jets launched by rotating black holes are powerful emitters of non-thermal radiation. Extraction of the rotational energy via electromagnetic stresses produces magnetically-dominated jets, which may become turbulent. Studies of magnetically-dominated plasma turbulence from first principles show that most of the accelerated particles have small pitch angles, i.e. the particle velocity is nearly aligned with the local magnetic field. We examine synchrotron-self-Compton radiation from anisotropic particles in the fast cooling regime. The small pitch angles reduce the synchrotron cooling rate and promote the role of inverse Compton (IC) cooling, which can occur in two different regimes. In the Thomson regime, both synchrotron and IC components have soft spectra, $ u F_ upropto u^{1/2}$. In the Klein-Nishina regime, synchrotron radiation has a hard spectrum, typically $ u F_ upropto u$, over a broad range of frequencies. Our results have implications for the modelling of BL Lacs and Gamma-Ray Bursts (GRBs). BL Lacs produce soft synchrotron and IC spectra, as expected when Klein-Nishina effects are minor. The observed synchrotron and IC luminosities are typically comparable, which indicates a moderate anisotropy with pitch angles $thetagtrsim0.1$. Rare orphan gamma-ray flares may be produced when $thetall0.1$. The hard spectra of GRBs may be consistent with synchrotron radiation when the emitting particles are IC cooling in the Klein-Nishina regime, as expected for pitch angles $thetasim0.1$. Blazar and GRB spectra can be explained by turbulent jets with a similar electron plasma magnetisation parameter, $sigma_{rm e}sim10^4$, which for electron-proton plasmas corresponds to an overall magnetisation $sigma=(m_{rm e}/m_{rm p})sigma_{rm e}sim10$.
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