A hadith has been narrated about using the siwak crosswise, but I do not take a hadith like it as evidence.
Bahz narrates that the Messenger of Allah (peace and blessings be upon him) used to use the siwak crosswise like this; it is like this in this narration.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 173