Can I store half of a record on one page and half of the record on another page?
You are splitting hairs over semantics. A physical record cannot span pages. A physical record has to fit in a page. 

But a index (or heap) row consists potentially from more than one physical records. An index row will consist from exactly one record in the IN_ROW_DATA allocation unit and zero, one or more records in ROW_OVEFLOW_DATA and/or LOB_DATA. The IN_ROW_DATA record contains pointers to overflow or LOB records. These off-row records themselves may contain more pointers to form a chain or a tree (this is how a VARCHAR(MAX) column can have  values that are obviously much larger than 8060 bytes). Such a large value will consist from many individual records, each of 8060 bytes or less. None of these record will ever span multiple pages.

And I did not even touch columnstore index rows, which are completely different.