Early reports from oral argument suggest that Justice Scalia was intensely skeptical of the claims of anonymity for ballot measure signers in Doe v. Reed. The transcript is not out yet, but the reports suggest that Justice Scalia echoed the sentiments he expressed in his dissent in the 1995 case, McIntyre v. Ohio Elections Commission:

I do not know where the Court derives its perception that “anonymous pamphleteering is not a pernicious, fraudulent practice, but an honorable tradition of advocacy and of dissent.” Ante, at 23. I can imagine no reason why an anonymous leaflet is any more honorable, as a general matter, than an anonymous phone call or an anonymous letter. It facilitates wrong by eliminating accountability, which is ordinarily the very purpose of the anonymity. There are of course exceptions, and where anonymity is needed to avoid “threats, harassment, or reprisals” the First Amendment will require an exemption from the Ohio law. Cf. NAACP v. Alabama ex rel. Patterson, 357 U.S. 449 (1958). But to strike down the Ohio law in its general application–and similar laws of 48 other States and the Federal Government–on the ground that all anonymous communication is in our society traditionally sacrosanct, seems to me a distortion of the past that will lead to a coarsening of the future.

Given this view, it is hard to see why the Court granted a stay of the lower court judgment and granted cert.
Justice Stevens, the author of McIntyre, dissented from the stay order. Presumably, Justices Breyer, Ginsburg, and Sotomayor are going to be skeptical of these claims of anonymity, even if they did not join Justice Stevens’ dissent from granting the stay. Did Justice Scalia vote for the stay and cert? Perhaps he did so to see if there was any real evidence of threats, harassment, etc. Once he was convinced that there was no evidence of such threats, he stood ready to reject Jim Bopp’s pleas for anonymity.
Assuming the Chief Justice agrees with Justice Alito that there is a right to anonymity in these cases, then Justice Stevens could be the one assigning a majority opinion, and he could assign that opinion to Scalia. I’m sure Justice Stevens will enjoy being on the right side of Justice Scalia’s ire as he prepares to retire from the Court.
It makes me wonder if Justice Alito (or the Chief—likely not Justice Thomas, who tangled in McIntyre with Justice Scalia over this issue) miscalculated in pushing for the Court to hear this case.