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
(a) It is narrated from Sayyiduna Rabi'ah bin Aktham (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) used to use the siwak crosswise and drink by sipping, and he would say: "This is more wholesome, more pleasant and more healthful."
(b) In the narration of Bahz there are these additional words: that he would breathe three times and say: "This is more wholesome, more pleasant and more healthful."
(b) Bahz is known only through this hadith. As for Sayyiduna Rabi'ah bin Aktham (may Allah be pleased with him), he was martyred at Khaybar.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 174
Hadith Gradingمحدثین:ضعيف
Hadith Takhrij«ضعيف۔ أخرجه العقيلي في الصعفاء 3/229»
It is narrated from Ata bin Abi Rabah (may Allah have mercy on him) that the Messenger of Allah (peace and blessings be upon him) said: "When you drink, drink by sipping, and when you use the siwak, use it crosswise."
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الطهارة / 175
Hadith Gradingمحدثین:ضعيف أخرجه اوب داؤد في المراسيل