Rotationのアニメーション版で、一定期間にわたって回転が徐々に変化する動作を表現することができます。
import 'package:flutter/material.dart';

void main() {
  runApp(const MyApp());
}

class MyApp extends StatelessWidget {
  const MyApp({Key? key}) : super(key: key);

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Demo',
      theme: ThemeData(
        primarySwatch: Colors.blue,
      ),
      home: const MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  const MyHomePage({Key? key, required this.title}) : super(key: key);

  final String title;

  @override
  State<MyHomePage> createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  double turns = 0.0;

  void _changeRotation() {
    setState(() => turns += 1.0 / 8.0);
  }

  @override
  Widget build(BuildContext context) {
    return Column(
      mainAxisAlignment: MainAxisAlignment.center,
      children: <Widget>[
        ElevatedButton(
          child: const Text('Rotate Logo'),
          onPressed: _changeRotation,
        ),
        Padding(
          padding: const EdgeInsets.all(50),
          child: AnimatedRotation(
            turns: turns,
            duration: const Duration(seconds: 1),
            child: const FlutterLogo(),
          ),
        ),
      ],
    );
  }
}