I'm going to skip the science lesson, and go straight to the solution here! If you're seeing unwanted orange or yellow tones in your hair, here's my advice:
USE A COLOUR DEPOSITING SHAMPOO & MASK
Violet or blue shampoo and masks deposit colour onto the surface of your hair, keeping the tones of your hair nice and cool. We're going for the opposite colour on the colour wheel here: so if you're looking to banish orange you'll need a blue range, or if you're looking to cancel out yellow go with a violet range.
I stock two different ranges of colour depositing haircare at Tint Department: Fanola and 18 in 1 Intense. Both are amazing, but here's some info to help you choose. Fanola No Yellow & No Orange are insanely powerful colour depositors, but the shampoo does contain sulphates so you make notice some drying. The results you've seen all over social media really do speak for themselves though!
18 in 1, on the other hand, does not contain sulphates - that's the major difference. It's also packed with insanely powerful pigments (wear gloves caus it can stain hands!), but also full of moisturising properties so you'll be left with soft, toned hair.
STICK WITH ASH BASED COLOURS AND TONERS
Hair colour comes in many different undertones, and if you're looking to banish brassiness anything in an Ash is your new BFF. These, as the name suggests, are on the ashy or cool-toned end of the colour spectrum - the perfect antidote to unwanted warm tones!
If you're in the process of lightening, or have just bleached your hair, my suggestion will always be to go with a demi-permanent colour. Demi colours have just as much toning power as permanent colours, but are waaayyy less damaging. They're essentially "staining" the hair and will last for up to 24 washes. If you're rocking a brilliant shade of orange, try out the Wella Demi in 7A - this will tone down the orange and leave you with a light ashy bronde (brown-blonde). Looking to get rid of yellow tones instead? Try out the Wella Demi in 10NA. This will leave you with a super light ash blonde, very similar to the famous Wella Colour Charm in T18, but without risking your hair to any more damage.