The Type Ia supernova (SN Ia) SN 2000cx was one of the most peculiar transients ever discovered, with a rise to maximum brightness typical of a SN Ia, but a slower decline and a higher photospheric temperature. 13 yr later SN 2013bh (also known as iPTF13abc), a near identical twin, was discovered and we obtained optical and near-infrared photometry and low-resolution optical spectroscopy from discovery until about 1 month past r-band maximum brightness. The spectra of both objects show iron-group elements [Co II, Ni II, Fe II, Fe III and high- velocity features (HVFs) of Ti II], intermediate-mass elements (Si II, Si III and S II) and separate normal velocity features (̃12 000 km s$^-1$) and HVFs (̃24 000 km s$^-1$) of Ca II. Persistent absorption from Fe III and Si III, along with the colour evolution, implies high blackbody temperatures for SNe 2013bh and 2000cx (̃12 000 K). Both objects lack narrow Na I D absorption and exploded in the outskirts of their hosts, indicating that the SN environments were relatively free of interstellar or circumstellar material and may imply that the progenitors came from a relatively old and low-metallicity stellar population. Models of SN 2000cx, seemingly applicable to SN 2013bh, imply the production of up to ̃1 M$_☉$ of $^56$Ni and (4.3-5.5) × 10$^-3$ M$_☉$ of fast-moving Ca ejecta.