A diagonal relationship exists between certain pairs of diagonally adjacent elements in the periodic table's second and third periods (first 20 elements).
These pairs are seen as lithium (Li) and magnesium (Mg), beryllium (Be) and aluminum (Al), boron (B) and silicon (Si), and many more.
On both of the descending groups and crossing by one element, the changes in them cancel each other out.
On the other hand, elements with similar properties with similar chemistry are often found - the atomic size, electro negativity, and properties of compounds of the diagonal members are said to be similar.
The size decreases from left to right and increases going down the periodic table.
So, going one element right and one element down would give in a similar size for the two elements diagonal to each other.
The ionization energies will be the same for the diagonal elements since the periodic trends also oppose each other.
The electron affinities are harder to guess, but the atoms with similar size and ionization energy should have similar electron affinities.
Therefore, the pairs of elements have similar sizes and electronegativity, resulting in similar properties.