Although the Tudor Rose is not a real flower, the flowers that were united are.
The Red Rose of Lancaster is also called the Apothecary’s Rose, Old Red Damask, the French Rose, and the Rose of Provinces. It's scientific name is Rosa Gallica Officinalis. It is a small shrub with dark pink, fragrant, semi-double flowers. It is possibly the first cultivated rose. It was discovered by ancient Persians and Egyptians while the rose grew wild in central Asia. The Greeks and Romans were the first to grow them; later, the French and Dutch favored it. The flower was used for perfume and for medicinal purposes. The plant prefers a cool enviroment and it's flowers bloom once per season.
The White Rose of York (Rosa Alba) is said to have been first adopted as the emblem of York by Edmund of Langley, the first Duke of York in the 14th century. It symbolizes purity and the Virgin Mary. It is a vigorous plant of 5 to 7 feet tall with singular white, flat flowers, each with five petals. Today, the white flower is the emblem of Yorkshire.
It may also be worth noting Rosa Mundi, a sport (a spontanious genetic mutation) of Rosa Gallica. It was named after Rosamund de Clifford, mistress of Henry II. It is similar to Rosa Gallica aside from it's red and pink striped flowers. Don't confuse this with the Tudor Rose. This rose was created and accounted for centuries before the War of the Roses.