The common way to start a 2x2 solve is by building a face or layer. To take this further, you can pseudo build so that they contain pieces from separate layers. At first it isn't that intuitive. A good strategy is to find a first pair and add to it. Below is a list of first pair types:

Some pairs equal another. Rotate the cube and its second identity is shown.

1=9, 2=10, 4=4, 5=5, 7=7, 11=11, and 12=12. The rest have no meaningful second form.

With the first pair chosen, there are several possible pair combinations to complete the face or layer. They are listed below. Parentheses around a number means that pair needs swapped. Parentheses around the plus sign means one of the two unlabled pairs needs swapped.

Scramble: R U' F' R' U' F2 U' F2 R

Layer: y x' F2 U' R

CLL: U L' U L F' U2 L2 F' L' U L

Scramble: F2 R' F R' F2 U R F R2

Layer: x y U B'

CLL: U2 L U2 L' U2 L' B L B' U' R'

Scramble: R2 F2 R U R U2 F2 R' U'

Face: y' x' U' R'

EG1: U R' U2 R' U2 R' U' R2 U' R

Scramble: U F R F2 U2 F' U2 F U2

Face: x' U2 L'

EG1: R' U2 R U' R2 U R' U R2

Scramble: U' F2 U' F2 U R F2 R'

Face: y x2 U' R2

EG1: R' F U2 R2 F R' U' F'

Scramble: U R2 F2 R' F2 U2 R' U'

Face: x' y F'

EG1: U' R2 U R' U2 R2 F R

Scramble: F2 R F' R' U F' R' U

Face: x U R2

EG2: U F R F' U R2 F' R U'

Scramble: U2 R F U' F R' U2 R2 U'

Face: x' F

EG2: U R2 B2 R' U2 R' U' R U' R' U R'

Layer: y x' F2 U' R

CLL: U L' U L F' U2 L2 F' L' U L

Scramble: F2 R' F R' F2 U R F R2

Layer: x y U B'

CLL: U2 L U2 L' U2 L' B L B' U' R'

Scramble: R2 F2 R U R U2 F2 R' U'

Face: y' x' U' R'

EG1: U R' U2 R' U2 R' U' R2 U' R

Scramble: U F R F2 U2 F' U2 F U2

Face: x' U2 L'

EG1: R' U2 R U' R2 U R' U R2

Scramble: U' F2 U' F2 U R F2 R'

Face: y x2 U' R2

EG1: R' F U2 R2 F R' U' F'

Scramble: U R2 F2 R' F2 U2 R' U'

Face: x' y F'

EG1: U' R2 U R' U2 R2 F R

Scramble: F2 R F' R' U F' R' U

Face: x U R2

EG2: U F R F' U R2 F' R U'

Scramble: U2 R F U' F R' U2 R2 U'

Face: x' F

EG2: U R2 B2 R' U2 R' U' R U' R' U R'