Is it normal to see periods during which my inverter feeds into and consumes from the public network at the same time, on my monitoring interface?

It’s perfectly normal. The IMEON web portal samples inverter’s performance data with an interval of 15 minutes. During these 15 minutes, it is likely that the solar inverter has successively fed power to the grid and consumed power from it, it is therefore not unusual that information concerning both, consumption from and injection to electricity grid are displayed together.