In fact base.Clone() returns an object of type equals to this.GetType(). A reflection invocation on the default constructor might be done in the DataGridViewTextBox.Clone() method.

It seems that the one methodology for implementing ICloneable would be:

public virtual object Clone()
{
   object clone = this.GetType().GetConstructor( Type.EmptyTypes ).Invoke( null );

   clone.MyData = this.MyData.Clone();
   clone.MyData1 = this.MyData1.Clone();
   ...
   return clone;
}

Then in the inherited class you'd override Clone like this:

public override object Clone()
{
   object clone = base.Clone();
   clone.MyOtherData = this.MyOtherData.Clone();
   return clone;
}