Warm Reds
Warm Reds are those that are biased towards orange. They include Cadmium Red Light, Cadmium Scarlet, Spectrum Scarlet, some brands of Cadmium Red Medium, Scarlet Lake, Organic Vermillion, Anthraquinoid Scarlet, Naphthol Red, Pyrrol Scarlet and many more. Some are opaque, some are transparent. It is useful to have one warm red and one cool red in a split primary palette. Pyrrol Scarlet, by Daniel Smith, is one of my favourites. (To see a huge range of warm, cool and mid reds, see my blog post here)
In mixing, a warm red makes bright oranges when mixed with a warm yellow, dull neutral purples with a warm blue, neutral browns when mixed with a cool blue and mid oranges when mixed with a cool yellow. Its opposite is turquoise, which, when mixed, will create a fascinating range of hues. (see below)
In mixing, a warm red makes bright oranges when mixed with a warm yellow, dull neutral purples with a warm blue, neutral browns when mixed with a cool blue and mid oranges when mixed with a cool yellow. Its opposite is turquoise, which, when mixed, will create a fascinating range of hues. (see below)
Pyrrol Scarlet

Pyrrol Scarlet is my favourite warm red. It is a gorgeous rich red colour that neutralises with Phthalo Blue GS to create a range of Indian red hues and a deep black. It is more transparent and cleaner than the Cadmium reds.
Cadmium Red Scarlet

Cadmium Red Scarlet. Daniel Smith watercolour.
Cadmium reds cover well since they are opaque. This is made with PR 108. Many companies make Cadmium Red in a light, medium and deep. Generally a 'light' will be more orange, a 'medium' a mid red and a 'deep' almost crimson. I prefer the transparent DS Pyrrol Scarlet, Organic Vermilion or Da Vinci Permanent Red or W&N Scarlet Lake as regular palette colours but there are times when the opacity of the cadmiums is just perfect in a painting.
Pure bright oranges - Scarlet Lake + Cadmium Yellow Medium

Scarlet Lake (left) mixed with Cadmium Yellow Medium (right)
With a warm red and a warm yellow, thee is no blue to neutralise or dull the oranges.
Pure bright oranges - Cadmium Red Medium + Cadmium Red Deep

Cadmium Red Medium (left) mixed with Cadmium Yellow Deep (right).
Neutrals and greys - Cadmium Red Medium + Phthalo Green

Cadmium Red Medium (left) mixed with Phthalo Green (right)
Red and green are opposites, and will neutralise eachother when mixed. Since Phthalo Green is the mixing opposite of a cool or crimson red, this combination will not produce blacks, but will made some great shadow colours.
Neutrals and greys - Cadmium Red Medium + Phthalo Turquiose

Cadmium Red Medium (left) mixed with Phthalo Turquoise - made from Phthalo blue and Phthalo Green - (right).
The exact mixing opposite of a warm red is turquoise, which can be made mixing Phthalo Blue and Phthalo Green (Phthalo Turquoise) or Ultrmarine Blue and Phthalo Green. (Ultramarine Turquoise). At full strength it creates a good black and a range of greys in dilution. It also creates brick reds and cool greys.