Answer: Denials of new trial motions are not appealable.
But these things are never quite that simple, are they?
Here are a few buts:
1. Denials of new trial motions are reviewable on appeal. This is expressed in the recent opinion in Leinen v. Carlton (D6 Jul. 30, 2021) no. H047030 (nonpub. opn.). The Walker v. Los Angeles County Metro. Transp. Auth. (2005) 35 Cal.4th 15 citation is the case on point: **
"Carlton filed a timely notice of appeal from the judgment entered on April 8, 2019. Carlton's challenge to the order denying his motion to continue trial is not a subject for direct appeal but is reviewable on an appeal from the judgment. (Freeman v. Sullivant (2011) 192 Cal.App.4th 523, 527.) His challenge to the denial of his motion for new trial is cognizable on appeal; although an order denying new trial is not itself appealable, it is reviewable from an appeal upon the underlying judgment. (Walker v. Los Angeles County Metro. Transp. Auth. (2005) 35 Cal.4th 15, 18.)"
2. Orders that only partially deny a new trial motion are appealable. That is because orders granting new trial motions are appealable under Code of Civil Procedure section 904.1(a)(4). So if you are unhappy with the trial court's denying part of your new trial motion, you have to appeal the order partially granting it. (See here for further discussion.)
3. Orders denying statutory motions to vacate and set aside a judgment (e.g., Code Civ. Proc., §§ 473, 663) are appealable. See Ryan v. Rosenfeld (2017) 3 Cal.5th 124. (This rule might not apply to nonstatutory motions.) But the appellant in the recent case of Maiden v. United Healthcare Services, Inc. (D4d3 Aug. 3, 2021) no. G058401 (nonpub. opn.) flubbed her notice of appeal. The appellant had filed both a motion for new trial and a motion to vacate. They were both denied, and the appellant took appeals from both denials. As you now know in light of points (1) and (2) above, the appellant's appeal from the denial of the new trial motion was ineffective. But the Court of Appeal still reviewed the order denying the motion to vacate (which it affirmed).
4. Finally, one recent unpublished opinion says: “An order denying a new trial is appealable as a postjudgment order under Code of Civil Procedure section 904.1, subdivision (a)(2).” But that is wrong. Ignore this. (See here.)
As a final tip: When preparing a motion for new trial, give a thought to whether a motion to vacate and set aside the judgment might be a better fit. While a motion for new trial is required to preserve certain issues for appeal (most commonly, excessive or inadequate damages and new evidence), the trial court has discretion to modify its statement of decision to correct any defects in the judgment you point out in your motion. By contrast, when you file a motion to vacate under Code of Civil Procedure section 663, the court has no such discretion. A good case on point is Hole v. Takekawa (1913) 165 Cal. 372, 375-76:
"[The statutes] do not authorize any change in any finding of fact. 'Section 663 of the Code Civil Procedure authorizes simply the substitution of the judgment that should have been given as a matter of law upon the findings of fact in a case where the judgment already given is an incorrect conclusion from such findings. The court cannot on such a motion in any wav change any finding of fact."]. (See more here.)
Posttrial motions are an excellent time to consult an appellate attorney.