Merging time series dataframes in python

Question

I am working with some financial tick data. Given two example dataframes like this:

left_df =

Time                          Bid Price    Ask Price
2022-01-02 00:00:01.323597    100          101
2022-01-02 00:00:01.828502    100          101
2022-01-02 00:00:01.845020    100          101
2022-01-02 00:00:03.123567    100          101

right_df =

Time                          Bid Price    Ask Price
2022-01-02 00:00:01.110223    500          501
2022-01-02 00:00:01.800000    500          501
2022-01-02 00:00:03.100000    500          501

If I 'merge' left to right, I want the merged dataframe to look like this:

Time_left                     Time_right                    Bid Price_left    Ask Price_left    Bid Price_right    Ask Price_right
2022-01-02 00:00:01.323597    2022-01-02 00:00:01.110223    100               101               500                501
2022-01-02 00:00:01 828502    2022-01-02 00:00:01.800000    100               101               500                501                               
2022-01-02 00:00:01.845020    2022-01-02 00:00:01.800000    100               101               500                501
2022-01-02 00:00:03.123567    2022-01-02 00:00:03.100000    100               101               500                501

i.e. for each time_left x, get the most recent time_right y up to x, and y can be equal to x.

Whereas if I wanted to 'merge' right to left, the resulting dataframe should look like this:

Time_right                     Time_left                    Bid Price_right    Ask Price_right    Bid Price_left    Ask Price_left
2022-01-02 00:00:01.800000    2022-01-02 00:00:01.323597    500               501               100                101
2022-01-02 00:00:03.100000    2022-01-02 00:00:01.845020    500               501               100                101

What would be the most efficient way of doing this on a dataset that could potentially have tens of millions of rows?