If it's rolled up on a spool, how do they prevent reconstruction of the order in which votes were cast?
The rolls are not assessable to the poll workers and would be read only if there were a recount by the worker in the counting office. Each polling place has several voting machines. Also after you made all your choices it is printed on the paper for you to view, then you have the choice of changing if you made a mistake. There must be a process that marks these as not valid and then prints your new choices. In case of a recount these must be removed somehow. At my poll I go to any open machine. Other than these things I don't know what is done at the count office.
The rolls are not assessable to the poll workers and would be read only if there were a recount by the worker in the counting office. In a proper secret-ballot election, it's impossible for anyone to ever tell how a particular person voted unless the election was all but unanimous in a particular precinct (if there were 300 ballots cast in a precinct, 299 of them were for Joe Quimby, and one was for Lionel Hutz, the person who voted for Lionel Hutz would know that each of the other 299 ballots was cast by someone voting for Quimby).
If the ballots are stored, in order, on a paper roll there's no way a voter can be sure people won't discover how he voted. By contrast, if someone drops a card into a box which is large relative to the number of cards such that the card can fall randomly, then (especially if the box is turned over occasionally) it's impossible to associate any particular ballot with any particular voter.