Symmetrically is the adverb of symmetrical: it describes doing something, or something being placed, so that the result has matching halves. You arrange chairs symmetrically. Leaves grow symmetrically along a stem. A pattern repeats symmetrically across a cloth.

Because it is an adverb, it does a different job from the adjective, and this is worth feeling rather than memorising. The adjective tells you what a thing is: 'the room is symmetrical'. The adverb tells you how something was done or how things sit in relation to each other: 'the furniture is arranged symmetrically', 'the wings fold symmetrically'. If you can put 'in a matching, balanced way' in its place and the sentence still works, the adverb is right.

It shows up most in three kinds of writing. Description of design and architecture — how a garden, a facade, a stage is laid out. Science and maths — how a molecule is built, how forces are distributed, how a curve behaves symmetrically about a line. And instructions — fold the paper symmetrically, place the screws symmetrically around the plate. It is a long word and a slightly technical-sounding one, so in speech people usually say 'evenly', 'on both sides', or 'the same on each side'.

Though actually — evenly and symmetrically are not the same, and swapping them is the commonest real error. Evenly is about amount and spacing: spread the butter evenly, space the posts evenly. Symmetrically is about mirroring around a centre: five posts spaced evenly along a wall are even but not necessarily symmetrical, while two posts placed the same distance from the door on either side are symmetrical without anything being spread. When you mean 'the same amount everywhere', say evenly. When you mean 'balanced around a middle', say symmetrically.